The interaction of shock waves with rigid obstacles is of significant interest in aerodynamic science and other engineering applications. During the interaction of a shock wave with an obstacle, a very complex wave pattern which affects the shockwave induced flow is formed. The interaction process depends on a variety of physical parameters such as the shape of the obstacle, the shock wave strength and the type of gas in which the interaction occurs. In the present paper, the interaction of a planar shock wave with a cylinder and a sphere is investigated. Our investigation follows closely the recentwork of Sadot et al. [1] which dealt with shock tube experiments with low Mach number shocks, in the range 1.1 to 1.4. An empirical relation was proposed for the trajectory of the reflected wave. This relation was expressed in terms of non-dimensional distance and time and was shown to be applicable for the investigated range of Mach numbers, cylinder diameters and a general ideal gas. The purpose of the present work is to focus on the backward reflected wave, and in particular, on its velocity change as it progresses away from the leading edge of the cylinder/sphere. It is expected that the reflected wave initially propagates at the velocity of shock reflection from a rigid wall, and asymptotically decelerates to the velocity corresponding to that of a sonic wave in the shocked region. This theoretical behavior is born out by fine mesh hydro-code computations of the interaction problem. The paper is organized as follows: in Section 2 a theoretical background for the limiting velocities of the reflected shock is given, followed by a brief description of the numerical codes and the problem setup (Section 3). The results of simulations for various cases by different CFD codes are given in Section 4. We conclude (Section 5) with a summary and suggestion of future work.

The concept of passive shock deflectors investigated in this paper allows an easy entry of a shock wave into a shelter, while initiating further reflections and turbulent vortices by diffracting the transmitted shock over a single 90°-chevron facing the incoming flow. Gas dynamic aspects of different chevron placements that control the variation in the attenuation capacity were investigated experimentally, using pressure measurement and shadowgraph photography.

In light of recent terrorist attacks on facilities throughout the world, designers, planners, architects, and engineers are beginning to re-visit conventional approaches in the design of military and high-security facilities. Many existing buildings, structures and facilities should be better protected against man-made explosive hazards. For this reason the designs of public/private structures and facilities are getting much more attention in recent years. This paper presents the results of experimental and numerical investigations that were conducted in order to examine the capabilities of aluminum foams in mitigating the effect of blast waves acting on reinforced concrete (RC) beams and plates. Numerical simulations of the behavior of the investigated RC beams and plates under the dynamic loads support the experimental investigation results. The numerical simulations are based on finite element commercial and in-house codes. The experimental and numerical results for both the protected and unprotected RC beams and the conclusions obtained from these experiments and simulations are presented.

One of the main research goals of the Protective Technologies Research and Development Center at Ben Gurion University of the Negev is to develop reliable numerical codes capable of accurately simulating the behavior of structures protected with aluminum foams exposed to explosion-generated blast-wave loads. To achieve this objective, the mechanical properties of aluminum foams under dynamic loads must be known. Several types of experimental investigations were carried out to obtain the mechanical properties of aluminum foams at different strain rates. The tests were conducted using different types of loads: static tests using an Instron compressing machine, dynamic tests using an Instron compressing machine, impact tests using a 400-kg impact pendulum, and shock-wave impact tests using conventional shock tubes. Plots of the volumetric strain as a function of the pressure and the strain rate were obtained, and the numerical codes were calibrated and validated with the aid of the experimental results.

A numerical method has been described for the calculation of the unsteady flow of an incompressible viscous fluid in a distensible tapered tube possessing orthotropic viscoelastic properties. The formulation is quite general, including the fluid-wall interaction, the tube wall being characterized as a thin shell with negligible bending moments.Illustrative solutions have been presented for the biomedical phenomena of a pulsed flow of blood in an excised segment of human aorta, subjected to both radial and longitudinal deformations. The complete solution includes the time variations of wall shape, stresses, strains and fluid velocity, computed in conjunction with a realistic dynamic constitutive model for the aortic wall.The example presented possesses an impulsive inlet flow sufficient to produce "shock-like" signals along the vessel wall. These sharp wave fronts do not endanger the inherent stability of the numerical two-step Lax-Wendroff scheme, provided that the stability criteria are properly observed.

The oblique impact and penetration of long rods is investigated both experimentally and numerically. In the experiments a long copper rod impacted obliquely an aluminum plate at a velocity of 850 m/sec. We have chosen ductile materials like aluminum and copper to ensure that the results are not sensitive to the failure model used in the calculations. The low density of the aluminum, and the relatively small sizes of the target plates enabled us to resolve the material interfaces in the X-ray shadowgraph taken during the penetration. The numerical simulations were carried out in the multimaterial Euler-with-strength processor of the three dimensional code MSC/DYTRAN. There is a good agreement between the computations and the experimental results, as to the shape of the projectile and target during the penetration process.

Introduction I stationary supersonic flow over a sharp wedge of halfangle 6, an oblique shock is attached to the leading edge, deflecting the flow velocity into an orientation parallel to the wedge surface (Sec. 117 of Ref. 1), provided 5 \. This ratio was shown to be a function of the wedge angle 5, the ratio of specific heats y (assuming a perfect gas) and M\. Further examination of the shock formation process leads to the conclusion that a more accurate model can be devised, still in the realm of simple analysis. The self-similar impulsively started flow near a wedge of finite half-angle was described qualitatively by Sakurai. He pointed out that when the flow commences abruptly (either by setting the fluid in motion or by impulsive acceleration of the wedge), the self-similar shock formed about the wedge consists of three segments (Fig. 1): 1) a plane oblique shock attached to the leading edge, having uniform intensity; 2) a curved transient shock of nonuniform intensity; and 3) a plane uniform stopping shock parallel to the wedge surface. In regions 1 and 3 the flow is uniform, and in region 2 it is nonuniform. No solutions exist for this nonuniform region. The present study formulates an alternative, more accurate model for the rate of formation of the oblique shock. The accuracy of the model is checked by comparison with numerical computations, which, in the absence of experimental data, are our sole guideline.

The nonlinear elastic response of large arteries subjected to finite deformations due to action of biaxial principal stresses, is described by simple constitutive equations. Generalized measures of strain and stress are introduced to account for material nonlinearity. This also ensures the existence of a strain energy density function. The orthotropic elastic response is described via quasi-linear relations between strains and stresses. One nonlinear parameter which defines the measures of strain and stress, and three elastic moduli are assumed to be constants. The lateral strain parameters (equivalent to Poisson's ratios in infinitesimal deformations) are deformation dependent. This dependence is defined by empirical relations developed via the incompressibility condition, and by the introduction of a fifth material parameter. The resulting constitutive model compares well with biaxial experimental data of canine carotid arteries.

The phenomenon of high-amplitude inflation waves resulting from a sharp axial acceleration of the aorta, as may occur in road accidents, is investigated theoretically. The aorta is modeled as an axisymmetric tapered membranic shell (tube) made of an incompressible, nonlinear viscoelastic material with cylindrical orthotropy. It is filled with an inviscid, incompressible fluid whose flow is considered as quasi-one dimensional along the tube axis. The equations of motion of the tube and of the fluid are solved numerically, by using a two-step explicit scheme, for several axial acceleration profiles. The solutions shows that an inflation wave is generated and it propagates in opposite direction to that of the acceleration. The wall stresses, deformations and their time derivatives as well as fluid velocity and pressure are determined along the tube at different time intervals. Peak axial and circumferential stresses are high, with the latter far exceeding the former. These stresses may cause rupture of the aorta.

A “channel” model was developed for the purpose of simulating the interactive fluid-structural response of curved pipes to pressure pulses. Simulation is shown to have been achieved analytically in both the axisymmetric (“breathing”) and transverse (“bending”) modes of interactive behavior.

A mathematical model for large amplitude wave propagation in a thin walled distensible tube is developed. The tube wall is considered as a membranic shell made of an incompressible, non-linear viscoelastic material with cylindrical orthotropy. The fluid is regarded as incompressible and inviscid and the flow is quasi-one-dimensional. The case of a pressure step applied at one end of a uniform straight tube is solved as an example. The system of partial differential equations, describing the motions of the fluid and the wall, are integrated numerically by using a two-step explicit scheme. Flow and deformation variables as well as the wave velocity are determined in time and space.

A mathematical model for describing the interaction between a compressible fluid and an elastic shell is formulated as an initial boundary value problem. The partial differential equations of the model are discretized both in time and space by a finite-difference method. The stability of the resulting explicit difference schemes is analyzed by Kreiss' theory for the stability analysis of difference schemes in initial boundary value problems. It is shown that the stability properties of the schemes for the interaction problem may be influenced by the type of discretization in space used for the contact condition on the interface between the fluid and the shell and also by the approximation of the hydrodynamic pressure on the surface of the shell. A simple sufficient condition is found that will ensure the best possible stability properties of the schemes. Several of these, which are of practical interest, are analyzed.

Large amplitude waves may be generated in the aorta as a result of body impacts, as in traffic accidents. Such phenomenon is analogous, in some respects, to shock waves in a gas flowing through a rigid tube. Here the distensibility of the tube plays the role of the gas compressibility and the problem is one of interaction between the fluid and the wall. So far, most mathematical models that describe nonlinear large wave propagation in distensible tubes employ the method of characteristics. In this method the overall change across the wave can be computed directly with no need for details of the wave front. The constitutive relations of the wall material have been incorporated in this method as a tube cross section — pressure relation that neglects longitudinal stresses [l–7]. Another theory [8] for steady-state shock-structure, based on mathematical analogy with gas-dynamics shock waves, assumes the tube to be axially constrained (complete tethering). In the present study a mathematical model for large amplitude wave propagation is presented. Similar to the above methods the flow is assumed to be quasi-one dimensional. The blood vessel is treated as a membranic shell subjected to biaxial stresses. This model allows such effects as tube geometry, initial loading and boundary conditions to be accounted for as well as different constitutive relation for the wall material.

We present a new approach to the numerical simulations of jet target interactions. We represent the jet by sources of mass, momentum and energy moving with a prescribed velocity. This velocity is related to local flow parameters through the incompressible theory of jet penetration. A Lagrangian, 2-dimensional elastic plastic code is used to model the flow in the target. The interaction between the jet and the target is obtained by appropriate deposition of the jet material with its momentum and energy in the computational mesh of the target. To prevent large grid distortions the grid is continuously rezoned. For normal jet impingement the calculations are carried out with axial symmetry. Oblique incidence is treated approximately considering a “sheet jet” and a two dimensional plane strain flow. Results for normal and 45° oblique jet impacts on a 25.4mm thick homogeneous armor plate are presented. The computed and experimental distributions of behind-the-armor debris are compared. An additional case of a 45° oblique jet penetration into a multi-layered plate is carried out. The flow in the target and its influence on the penetration process is discussed.

A quasi one dimensional model for wave propagation in the aorta including initial tension is formulated and analyzed. Closed form expressions for maximal stretching are derived and wave profiles computed. Results are presented and a comparison is made with a completely tethered model.

The flow arising from an initial pressure discontinuity across a perturbed interface of two ideal gases is studied using analytical and numerical methods. In particular, the stability of the shock wave, the interface, and the rarefaction wave in the resulting flow are investigated. The equations of motion and the initial and boundary conditions are linearized for small perturbations, and a Fourier analysis is made in the lateral direction. The equations are then solved by the method of characteristics. The results show that the interface is unstable and its perturbations asymptotically acquire a constant rate of growth. The shock wave is stable and has rapidly damped oscillations, which appear to be unaffected by the instability of the interface.

The spectral stability theory of initial boundary value explicite finite-difference schemes is used to develop a stability analysis method for problems of fluid-structure interaction. By this analysis it is shown that due to the interaction between the structure and fluid stability restrictions on the time step may be more severe than commonly assumed. Four schemes of practical interest are analyzed in detail. The validity of the stability analysis is tested by simulating the effects of underwater explosion on a submarine. The computational results corroborate the prediction of the analysis concerning the stability boundary.