The flow field near the triple point in steady shock reflection

This paper investigates the flow field near three intersecting shock waves appearing in steady Mach reflection. Results of numerical computations reveal a “von Neumann Paradox”—like feature for weak shock waves, in which the flow field between the reflected and the Mach stem is smooth with no distinct slip flow region and changes rather smoothly. An analytical solution of the Navier–Stokes equations constructed using a polar–coordinate system gives a flow field with the same properties as the numerical simulation.     

Numerical simulation of Mach reflections

In this paper, the “FLIC” difference method with triangular mesh is adopted to numerically simulate the regular and Mach reflections that occur when a shock wave pass around a wedge. The compuational result is compared with the shock tube experimental results of G. Ben-Dor and I. I. Glass. The comparison shows that the position, shape of shock wave and height of Mach stem all show a good agreement. Consequently, the “FLIC” difference method with triangular mesh is quite satisfactory in numerical simulation of the regular and Mach reflections.     

A Novel “Subset Splitting” Procedure for Digital Image Correlation on Discontinuous Displacement Fields

Digital Image Correlation (DIC) is an easy to use yet powerful approach to measure displacement and strain fields. While the method is robust and accurate for a variety of applications, standard DIC returns large error and poor correlation quality near displacement discontinuities such as cracks or shear bands. This occurs because the subsets used for correlation can only capture continuous deformations from the reference to the deformed image. As a result the regions around discontinuities are typically removed from the area of interest, before or after analysis. Here, a novel approach is proposed which enables the subset to split in two sections when a discontinuity is detected. This method enables the measurement of “displacement jumps”, and also of displacements and strains right by the discontinuity (for example a crack profile or residual strains in the wake). The method is validated on digitally created images based on mode I and mode II asymptotic displacement fields, for both sub-pixel and super-pixel crack opening displacements. Finally, an actual fracture experiment on a high density polyethylene (HDPE) specimen demonstrates the robustness of the method on actual images. Compared to other methods capable of handling discontinuities, this novel “subset-splitting” procedure offers the advantage of being a direct extension of the now popular standard DIC, and can therefore be implemented as an “upgrade” to that method.     

Flow visualization using fluorescence from locally seeded I2 excited by an ArF excimer laser

 A new concept for flow visualization is demonstrated in which fluorescence from locally seeded iodine is viewed in the wake of simple aerodynamic models at Mach 6. Localized seeding is performed by painting a small area of a ceramic model with a tincture of iodine. When the model is injected into the flow, the adsorbed iodine is entrained into the boundary layer, follows the model contour, and ultimately mixes into the wake region. Planar “snapshots” of the wake flow are taken by exciting the iodine with an ArF excimer laser sheet at 193 nm and observing the fluorescence in the 210–600 nm region with an intensified CCD camera. Received: 17 July 1997/Accepted: 12 August 1998     

Effect of branching in cross-slot flow: the formation of “W cusps”

The sensitivity of the principal stress difference (PSD) profile to molecular architecture is demonstrated for flow in a cross-slot geometry. For materials with low levels of branching, the pattern along the outlet centre line exhibited “single cusps”, while an increase in molecular branching was found to lead to “W cusps”. The formation of these W cusps was found to be independent of extensional rate for the conditions probed, and they were formed initially at the stagnation point and travelled along the outlet centre line with time. Comparison with simulations performed using a multi-mode “pom-pom” model failed to predict W cusps, although the general level of PSD was accurately captured.     

Effects of the weak/micro-discontinuity of interface on the fracture behavior of a functionally graded coating with an inclined crack

The mechanical model was established for the anti-plane fracture problem of a functionally graded coating–substrate system with a coating crack inclined to the weak/micro-discontinuous interface. The Cauchy singular integral equation for the crack was derived using Fourier integral transform, and the Lobatto–Chebyshev collocation method put up by Erdogan and Gupta was used to get its numerical solution. Finally, the effects of the weak/micro-discontinuity of the interface on SIFs were analyzed, the “affected regions” corresponding to the two crack tips have been obtained and their engineering significance was discussed. It was indicated that, for the crack tip in the corresponding “affected region”, to reduce the weak-discontinuity of the interface and to make the interface micro-discontinuous are the two effective ways to reduce the SIF, and the latter way always has more remarkable SIF-reduction effect. For the crack tip outside the “affected region”, its SIF is mainly influenced by material stiffness, and to prevent such a tip from growing toward the interface “softer coating and stiffer substrate” is a more advantageous combination than “stiffer coating and softer substrate”.     

Level set method for numerical simulation of a cavitation bubble,its growth,collapse and rebound near a rigid wall

A level set method of non-uniform grids is used to simulate the whole evolution of a cavitation bubble, including its growth, collapse and rebound near a rigid wall. Single-phase Navier–Stokes equation in the liquid region is solved by MAC projection algorithm combined with second-order ENO scheme for the advection terms. The moving interface is captured by the level set function, and the interface velocity is resolved by “one-side” velocity extension from the liquid region to the bubble region, complementing the second-order weighted least squares method across the interface and projection inside bubble. The use of non-uniform grid overcomes the difficulty caused by the large computational domain and very small bubble size. The computation is very stable without suffering from large flow-field gradients, and the results are in good agreements with other studies. The bubble interface kinematics, dynamics and its effect on the wall are highlighted, which shows that the code can effectively capture the “shock wave”-like pressure and velocity at jet impact, toroidal bubble, and complicated pressure structure with peak, plateau and valley in the later stage of bubble oscillating. The project supported by the National Natural Science Foundation of China (10272032 and 10672043). The English text was polished by Keren Wang.     

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     

Treatment of interface problems with Godunov-type schemes

A correction is proposed of Godunov-type schemes, yielding a perfect capture of contact discontinuities in hydrodynamic flows. The correction method is based upon the following simple idea: If an Euler scheme is employed starting from a non-degraded solution at a certain instant of time, the presence of a discontinuity will entail, at the next instant, the degradation of the solution at the two points adjacent to the discontinuity only. On the other hand, an exact solution of the Riemann problem yields the state variables at the nodes affected by numerical diffusion can be corrected. The method is applied to problems involving a gas-liquid interface. The liquid is supposed to be compressible, obeying an equation of state of the “Stiffened Gas” type, for which a solution to Riemann's problem is readily obtained.     

Transient heat conduction in a solid slab using multiple-scale analysis

In this paper we study the unsteady heat conduction due to a sudden temperature step in the external surfaces of a solid slab. In order to estimate the temperature profile in the solid, we applied the multiple-scale perturbation technique by identifying the “early” and “late” transient regimes for small values of the Biot number, Bi. In this sense, we have re-visited the classical lumped method, incorporating this particular case as an asymptotic limit, which is fully described by the “late” regime for small values of Bi. Once the temperature distribution is analytically predicted, this solution is compared against the exact solution and with other analytical results obtained by using regular perturbation techniques, for different values of the Biot number Bi. Observing a good agreement between the corresponding comparisons, we obtain a very simple and useful formula to predict the nondimensional temperature of the solid slab.     

Ideal incompressible flow around a wedge tip

A planar analog of conical flows is considered: an inviscid incompressible fluid flow around a wedge tip. A class of conical flows is found where vorticity is transported along streamlines by the potential component of velocity. Problems of a wave “locked” in the corner region and of a flow accelerating along the rib of a dihedral angle are considered. By analogy with an axisymmetric quasi-conical flow, a planar quasi-conical flow of the fluid is determined, namely, the flow inside and outside the region bounded by tangent curves described by a power law. Conditions are found where vorticity and swirl produce a significant effect. An approximate solution of the problem of the fluid flow inside a “zero” angle is obtained. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 57–65, March–April, 2007.     

Two-group kinetic theory of vehicular traffic

Summary We study a two-group model of vebicular traffic oli a highway which generalizes the Prigogine-Herman model. According to the two-group theory, drivers are divided into “slow” and “fast”. Correspondingly, we introduce a slow and a fast desired distribution representing the program that drivers of each of the two groups wish to follow. We first prove the existence and the uniqueness of a solution belonging to a suitable Banach space by using the theory of semigroups. Then, we indicate some sufficient conditions for the stability of an x and t independent solution.

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Sommario Si generalizza il modello di Prigogine-Herman del traffico automobilistico su un'autostrada, introducendo due gruppi di guidatori: i “lenti” ed i “veloci”. Ognuno dei due gruppi riene caratterizzato da un'opportuna funzione “desiderio”, che traduce i programmi che i guidatori intendono seguire nel corso del loro viaggio. Facendo poi uso della teoria dei semigruppi, si prova l'esistenza e l'unicità di una soluzione appartenente ad un opportuno spazio di Banach e si danno condizioni sufficienti per la stabilità.

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Work performed under the auspices of the National Research Council (C.N.R., Gruppo Nazionale per la Fisica Matematica).As the choice of the appropriate model for these problems is still a controversial matter, the Editorial Board welcomes any comment or discussion from the readers.     

Quantitative schlieren measurements of coherent structures in a cavity shear layer

Quantitative flow-field data were obtained in a planar shear layer spanning an open cavity with an extension of the schlieren method. The technique is based on the measurement of light-intensity fluctuations in a real-time schlieren image. Data were collected using a fiber-optic sensor embedded in the imaging screen coupled to a photodetector. Time-resolved measurements of the instantaneous density gradient at a point in the two dimensional flow cross section were thus obtained. Detailed surveys were carried out with both the optical instrument as well as a hot wire at a Mach number of 0.25 and with the optical instrument alone at a Mach number of 0.6. A comparison of the results shows that the non-intrusive technique can accurately measure the growth rates of instability waves in the initial “linear” region of the shear layer. The density-gradient fluctuations measured at different locations (and times) were synchronized by using a microphone inside the cavity as a reference and integrated to yield profiles of the density fluctuations associated with the dominant large-scale structures in the shear layer. Such quantitative visualization is expected to clarify the mechanism of sound generation by shear-layer impingement at the cavity trailing edge and elucidate the nature of this sound source. Received: 28 December 1999/Accepted: 15 March 2000     

Flow measurement on an oscillating pipe flow near the entrance using the UVP method

 The authors have carried out a study to investigate and clarify the characteristics of purely oscillating pipe flows over the developing region. The main objective of this study is to establish the method of time-dependent velocity profiles obtained by the ultrasonic velocity profile (UVP) measurement method. First, the relationship between the test fluids and the microparticles, as reflectors of ultrasonic pulses, was investigated. In addition, the relationship between the sound speeds of the test fluids and the wall materials was studied. Second, the UVP was used to obtain the instantaneous velocity profiles in oscillating pipe flows, and the developing characteristics of the flows were analyzed. Finally, the “entrance length” (by analogy with a unidirectional pipe flow) required for oscillating pipe flows was analyzed by examining the amplitude of the harmonic spectral components of the oscillating frequency. A fast Fourier transform (FFT) is proposed as the applicable method to estimate the “entrance length”. From the Fourier transform of the velocity on the centerline, nonlinear oscillation of fluid occurs in the “entrance length” of the oscillating flows, and the viscous dissipation of the higher-order velocity harmoncis determines the entrance region. The “entrance length” can be obtained from the dissipation length of the third-order harmonic. These results prove that the UVP method is highly applicable to carry out the flow measurement in the “entrance length” of oscillating pipe flow. Received: 20 March 2000 / Accepted: 10 August 2001     

Remarks on “Floating Bodies in Neutral Equilibrium”

This note offers historical comments on mathematical life in the city of Lwów, and points out that the paper cited in the title contains a solution of Problem 19 in “The Scottish Book”, prepared just prior to World War II in the “Scottish Café” located in that city.     

Interference of stationary and non-stationary shock waves

The mathematical model of a gasdynamic discontinuity is used in the area of study concerning gas flows with large gradients of gasdynamic functions. Gasdynamic functions before and after the discontinuity meet non-linear algebraic equations called the dynamic compatibility conditions on the discontinuities. Different modes of shock wave structures forming as a result of regular or irregular interference of the incoming discontinuities of different types are described. Ranges of the initial flow parameters definition such that either shock wave structures of different modes take place or interference equations have no solutions are determined. Most attention is given to arbitrary triple shock-wave configurations. Their classification is proposed. Differential characteristics of the steady flow are studied. The notion “differential characteristics” includes first derivatives of the fundamental gasdynamic parameters with respect to natural coordinates and curvatures of the discontinuities surfaces. Effect of unsteadiness on the triple-shock configuration is examined. Some problems arising at creation of complete local theory of steady and propagating gasdynamic discontinuities interference are formulated.     

Measure-theoretic foundations of statistical mechanics

The basic formulas of classical equilibrium statistical mechanics are derived from well-known theorems in measure theory and ergodic theory. The method used is a generalization of the methods of Khinchin and Grad and deals with several, in fact a “complete set”, of “invariants” or “integrals of the motion”. Most of the results are simple corollaries of Birkhoff's ergodic theorem, and since time-averages are used, the whole approach is characterized by an absence of statistical “ensembles” and probability notions. In the course of the development a “generalized temperature” is introduced, and a generalization of the second law of thermodynamics is derived. Formulas for the “microcanonical”, “canonical”, and “grand canonical” distributions appear as special cases of the general theory.     

Amorphous and crystalline states of ultrasoft colloids: a molecular dynamics study

In this work, we study the temperature-induced development of “dynamically arrested” states in dense suspensions of “soft colloids” (multi-arm star polymers and/or block-copolymers micelles) by means of molecular dynamics (MD) simulations. Temperature increase in marginal solvents results in “soft sphere” swelling, dynamical arrest, and eventually crystallization. However, two distinct “dynamically arrested” states were found, one almost amorphous (“glassy”) and one with a considerable degree of crystallinity, yet lower than that of the fully equilibrated crystal. It is remarkable that even that latter state permitted self-diffusion in the timescale of the simulations, an effect that underlies the importance of the “ultra-soft” nature of inter-particle potential. The “number of connections” criterion for crystallinity proved to be very successful in identifying the ultimate thermodynamic trend from the very early stages of the α-relaxation. This paper was presented at the Third Annual Rheology Conference, AERC 2006, April 27–29, 2006, Crete, Greece.     

Analysis of the transient thermal field and heat flux in circular plates, heated rapidly by a fast-decaying magnetic field

The rapid heating of a circular conducting plate by a magnetic field decaying exponentially with time and its transition to a final steady-state is studied for the cases of both isolated and non-isolated plates. Analytic expressions are derived for the thermal field, the heat flux and the relaxation times. Both the ’‘thin” and the “thick” aspects of the problem are investigated. Emphasis is placed upon some characteristic parameters arising from the analytical solution. Attention is paid to the time constants, related to the combined (conduction and convection) thermal process. In fact, the ratio of these time constants determines the transition process up to the final steady-state of each region of the plate.     

A State-of-the-Knowledge Review on Pseudo-Steady Shock-Wave Reflections and their Transition Criteria

The distinguished philosopher Ernst Mach published the first known paper on the phenomenon of planar shock-wave reflections over straight wedges over 125 years ago in 1878. In his publication he presented two wave configurations that could result from this reflection process, a regular reflection (RR) and a configuration that was later named after him and called Mach reflection (MR) in the early 1940s. In 1945, Smith reported on an additional wave configuration, which had a reflected shock wave that was slightly different from that of the just-mentioned Mach reflection. Smith (OSRD Rep. 6271, Off. Sci. Res. Dev., 1945) did not ascribe any special importance to the wave configuration that he observed. The wave configuration that was observed and reported by Smith (OSRD Rep. 6271, Off. Sci. Res. Dev., 1945) was recognized as an independent one only about 5 years later when White (Tech. Rep. II-10, Princeton Univ. Dept. Phys., 1951) reported on the discovery of a new wave configuration that was named double-Mach reflections (DMR) because it had similar features to that of the Mach reflection wave configuration but all the features were doubled. For this reason the Mach reflection wave configuration has been re-named single-Mach reflection (SMR). (Until the late 1970s it was called simple-Mach reflection although nothing is simple about it.). The discovery of the double-Mach reflection revealed that the wave configuration that was first observed by Smith was an intermediate wave configuration between the SMR and the DMR wave configurations. For this reason it was named transitional-Mach reflection (TMR) (Until the early 1980s it was called complex-Mach reflection although it is not the most complex one.). Since the discovery of the DMR many investigations were aimed at elucidating the exact transition criteria between the above-mentioned four different wave configurations as well as some additional configurations and sub-configurations that were discovered later. In 1991 Ben-Dor published a monograph, entitled “Shock Wave Reflection Phenomena”, that was, in fact, a state-of-the-knowledge review of the phenomena. This state-of-the-knowledge will be referred to in the followings as the “old”-state-of-the-knowledge (This state-of-the-knowledge existed until the mid 1990s. A few years later Li and Ben-Dor (Shock Wave 5(1/2), 59–73, 1995) modified the analytical approach for evaluating the transition criteria from the single-Mach to the transitional- Mach reflection (SMR, ,TMR) and from the transitional-Mach to the double-Mach reflection (TMR, ,DMR) and presented some modified and new criteria for the formation and termination of both the TMR and DMR wave configurations. Experimental results from various sources revealed that the transition boundaries between the SMR, TMR and DMR wave configurations that were based on the modified analytical approach were better than those of the “old” state-of-the-knowledge that as mentioned earlier was summarized in Ben-Dor’s (Shock Wave Reflection Phenomena, Springer, 1991) monograph. Unfortunately, however, the results of Li and Ben-Dor’s (Shock Wave 5(1/2), 59–73, 1995) modified analytical approach have not been internalized, and publications by various scientists in the past decade neglected the revised and better transition criteria and kept on referring to the old and wrong criteria that appeared in Ben-Dor’s (Shock Wave Reflection Phenomena, Springer, 1991) monograph. For this reason, a state-of-the-knowledge review that is based on the above-mentioned 10-year-old findings of Li and Ben-Dor (Shock Wave 5(1/2), 59–73, 1995) is presented herein. At the first step, the “old” state-of-the-knowledge is presented.This paper was based on work that was presented at the 2nd International Symposium on Interdisciplinary Shock Wave Research, Sendai, Japan, 1–3 March 2005.