Numerical models for afterburning of TNT detonation products in air

Afterburning occurs when fuel-rich explosive detonation products react with oxygen in the surrounding atmosphere. This energy release can further contribute to the air blast, resulting in a more severe explosion hazard particularly in confined scenarios. The primary objective of this study was to investigate the influence of the products equation of state (EOS) on the prediction of the efficiency of trinitrotoluene (TNT) afterburning and the times of arrival of reverberating shock waves in a closed chamber. A new EOS is proposed, denoted the Afterburning (AB) EOS. This EOS employs the JWL EOS in the high pressure regime, transitioning to a Variable-Gamma (VG) EOS at lower pressures. Simulations of three TNT charges suspended in a $26\,\hbox {m}^3$ explosion chamber were performed. When compared to numerical results using existing methods, it was determined that the Afterburning EOS delays the shock arrival times giving better agreement with the experimental measurements in the early to mid time. In the late time, the Afterburning EOS roughly halved the error between the experimental measurements and results obtained using existing methods. Use of the Afterburning EOS for products with the Variable-Gamma EOS for the surrounding air further significantly improved results, both in the transient solution and the quasi-static pressure. This final combination of EOS and mixture model is recommended for future studies involving afterburning explosives, particularly those in partial and full confinement.     

The Γ-Limit of the Two-Dimensional Ohta–Kawasaki Energy. I. Droplet Density

This is the first in a series of two papers in which we derive a Γ-expansion for a two-dimensional non-local Ginzburg–Landau energy with Coulomb repulsion, also known as the Ohta–Kawasaki model, in connection with diblock copolymer systems. In that model, two phases appear, which interact via a nonlocal Coulomb type energy. We focus on the regime where one of the phases has very small volume fraction, thus creating small “droplets” of the minority phase in a “sea” of the majority phase. In this paper we show that an appropriate setting for Γ-convergence in the considered parameter regime is via weak convergence of the suitably normalized charge density in the sense of measures. We prove that, after a suitable rescaling, the Ohta–Kawasaki energy functional Γ-converges to a quadratic energy functional of the limit charge density generated by the screened Coulomb kernel. A consequence of our results is that minimizers (or almost minimizers) of the energy have droplets which are almost all asymptotically round, have the same radius and are uniformly distributed in the domain. The proof relies mainly on the analysis of the sharp interface version of the energy, with the connection to the original diffuse interface model obtained via matching upper and lower bounds for the energy. We thus also obtain an asymptotic characterization of the energy minimizers in the diffuse interface model.     

Hybrid schemes to compute contact discontinuities in Euler equations with any EOSSchémas hybrides pour la simulation des discontinuités de contact des équations d'Euler avec une loi d'état quelconque

We propose here some explicit hybrid schemes which enable accurate computation of Euler equations with arbitrary (analytic or tabulated) equation of state (EOS). The method is valid for the exact Godunov scheme and some approximate Godunov schemes. To cite this article: T. Gallouët et al., C. R. Mecanique 330 (2002) 445–450.     

Une modélisation thermomécanique unidimensionnelle de la propagation d'un front de changement de phase dans un monocristal d'AMFA one-dimensional thermomechanical modeling of phase change front propagation in a SMA monocrystal

We propose a behavioral modeling that takes into account the thermomechanical couplings accompanying the phase transition in single-crystal CuZnAl samples. The goal of this model is to put forward the significant role played by the heat diffusion in the propagation mode of the phase change fronts. Numerical simulations showed the existence of such a phase change front and predicted the calorimetric and kinematic effects accompanying its propagation. In particular, an inversion of the propagation way during a creep test and caused by an increase of the room temperature was correctly simulated by the model. To cite this article: A. Chrysochoos et al., C. R. Mecanique 331 (2003).     

Dynamic fluid–structure interaction of an elastic capsule in a viscous shear flow at moderate Reynolds number

The unsteady behavior of a 2-D circular elastic capsule was investigated in three viscous shear flows. An immersed boundary method (IBM) has been used to solve the dynamic fluid-structure interaction of the capsule. Computations were carried out in finite parameter ranges where the Reynolds number is Re=1-40 and the capillary number is Ca=0.0005-0.05, which is the ratio of the external viscous shear stress to the resistant elastic tensions of the membrane. For the simple shear flow, the effect of inertia on the transient behavior of the capsule was studied. For the pulsatile shear flow, two values of the peak fluid strain, T_f=1 and 5, were considered for the quasi-steady capsule mechanics. The capsule shows a cyclic structural response that includes subharmonics as the Reynolds number is elevated to 10 and 40. The capsule dynamic response includes a phase lag, which is a function of the capillary number, the Reynolds number, and the peak fluid strain. Finally, the capsule flowing in the Couette flow shows lateral migration due to the transient lift force, which is higher for lower Ca and higher Re. When capsules with diverse elasticity are dispersed along the velocity gradient, the capsule with a hard membrane experienced greater lift than the one with a soft membrane.     

Flow around four cylinders arranged in a square configuration

This paper presents an experimental study of the flow around four circular cylinders arranged in a square configuration. The Reynolds number was fixed at Re=8000, the pitch-to-diameter ratio between adjacent cylinders was varied from P/D=2 to 5 and the incidence angle was changed from α=0° (in-line square configuration) to 45° (diamond configuration) at an interval of 7.5°. The flow field was measured using digital Particle Image Velocimetry (PIV) to examine the vortex shedding characteristics of the cylinders, together with direct measurement of fluid dynamic forces (lift and drag) on each cylinder using a piezoelectric load cell. Depending on the pitch ratio, the flow could be broadly classified as shielding regime (P/D≤2), shear layer reattachment regime (2.5≤P/D≤3.5) and vortex impinging regime (P/D≥4). However, this classification is valid only in the case that the cylinder array is arranged nearly in-line with the free stream (α≈0°), because the flow is also sensitive to α. As α increases from 0° to 45°, each cylinder experiences a transition of vortex shedding pattern from a one-frequency mode to a two-frequency mode. The flow interference among the cylinders is complicated, which could be non-synchronous, quasi-periodic or synchronized with a definite phase relationship with other cylinders depending on the combined value of α and P/D. The change in vortex pattern is also reflected by some integral parameters of the flow such as force coefficients, power spectra and Strouhal numbers.     

The weak shear kinetic phase diagram for nematic polymers

We study the shear problem for nematic polymers as modeled by the molecular kinetic theory of Doi (1981), focusing on the anomalous slow flow regime. We provide the kinetic phase diagram of monodomain (MD) attractors and phase transitions vs normalized nematic concentration (N) and weak normalized shear rate (Peclet number, Pe). We then overlay all rheological features typically reported in experiments: alignment properties, normal stress differences and shear stress. These features play a critical role in the synthesis between theory and experiment for nematic polymers (Larson 1999; Doi and Edwards 1986). MD type is routinely used for rheological shear characterization: cf., flow-aligning 5CB (Mather et al. 1996a), tumbling PBT (Srinivasarao and Berry 1991), and 8CB (Mather et al. 1996b), evidence for a wagging regime (Mewis et al. 1997), out-of-plane kayaking modes (Larson and Ottinger 1991), and evidence for chaotic major director dynamics (Bandyopadhyay et al. 2000). MD transitions correlate with sign changes in normal stresses (Larson and Ottinger 1991; Magda et al. 1991; Kiss and Porter 1978, 1980). Furthermore, structure formation in shear devices appears to be correlated with monodomain precursor dynamics (Tan and Berry 2003; Forest et al. 2002a). In this paper we combine seminal kinetic theory results (Kuzuu and Doi 1983, 1984; Larson 1990; Larson and Ottinger 1991; Faraoni et al. 1999; Grosso et al. 2001), symmetry observations (Forest et al. 2002b), and mesoscopic results on the fate of orientational degeneracy in weak shear (Forest and Wang 2003; Forest et al. 2003a), together with our resolved numerical simulations, to provide the kinetic flow-phase diagram of Doi theory in the weak shear regime, 0<Pe<1, for infinitely thin rods. We report the "birth" of key rheological features at the onset of flow: sign changes and local maxima and minima in normal stress differences (N₁ and N₂) associated with MD transitions. These results serve as the basis for continuation of the kinetic phase diagram to Pe>1 ; as the definitive benchmark for any mesoscopic or continuum model; and experimental data can be compared in order to determine accuracy and limitations of the Doi theory in weak shear.     

Characterisation of gas-liquid two-phase flow in minichannels with co-flowing fluid injection inside the channel,part II: gas bubble and liquid slug lengths,film thickness,and void fraction within Taylor flow

Current research proofs the potential of apparatuses containing minichannel flow structures to intensify gas-liquid-solid contacting processes. The excellent heat and mass transfer in these devices as well as a sharp RTD mainly result from the Taylor flow regime. A proper design of corresponding contactors requires precise information on the provided interfacial areas. However, the characterisation of gas-liquid Taylor flow with industrially relevant fluids at elevated pressure and created by capillary injection devices gained little attention so far.This work analyses adiabatic gas-liquid Taylor flow in a square minichannel of 1.0 mm hydraulic diameter using water, water-glycerol, or water-ethanol mixtures as liquid phase and hydrogen or nitrogen as gas phase to cover a broad range of material parameters. In the mixing zone located within the flow channel, gas was injected into the co-flowing liquid by so-called capillary injectors with variable inner diameter (0.184, 0.317, 0.490 mm).Two different bubble forming mechanisms were identified leading to a complex interaction between physical properties of the fluids, geometrical parameters and the observed gas bubble and liquid slug lengths. According to the Pi-theorem, these lengths were affected by 6 dimensionless groups, namely (u_G,s/ u_L,s), Re_L, We_L, (d_In,CI/ d_h), (d_Ou,CI / d_h), and Θ^*. Based on more than 370 experimental data, novel correlations to predict gas bubble and liquid slug lengths were developed.     

A study on Brinkman number variation on water based nanofluid heat transfer in partially heated tubes

Even if the variation of Nusselt number with Reynolds number has been observed before, little to no studies are concerning nanofluids. The present numerical data in the laminar regime were also found to correlate well with the Brinkman number for the individual sets and much better globally, by combining all the data sets for nanofluids and water.Thus, this work proposes a new correlation between heat transfer rate and Brinkman number which is a nondimensional number of viscosity, flow velocity and temperature. The results showed a good empirical equation that Nu/(Re^0.62 Pr^0.33) is dependent on a power law function to the Brinkman number in laminar flow regime. The proposed correlation can be applied both to water and nanofluid flow, with respect to the nanoparticle loading. It is expected that the equation suggested by this work can be useful to design heating/cooling devices.     

Development of empirical correlations to predict the secondary droplet size of impacting droplets onto heated surfaces

The present article reports an experimental analysis of the mechanisms of secondary atomization which occur at the impact of individual droplets onto heated targets. The experiments follow those reported in a previous article (Moreira et al. 2007) and encompass the use of different liquids and impact conditions. An image analysis system is combined with a phase Doppler interferometer to measure extended size distributions, which cover the full range of diameters generated at all heat transfer regimes. The results evidence that disintegration mechanisms depend on the heat transfer regimes; therefore, a universal relation cannot be devised for the outcome of droplet impact. Analysis shows that droplets impacting within the nucleate-boiling regime break-up by a thermal-induced mechanism associated with the vapour pressure at bubble nucleation sites, combined with liquid surface tension. On the other hand, within the film-boiling regime, disintegration is associated with radial disruption of the rim at the early instants after impact, as in non-heated targets, and with the rupture of the ligaments of the cellular structures. Functional relations available at the literature, mostly developed for impacts onto non-heated surfaces, are well fitted to the experimental results obtained within the film-boiling regime, since the break-up mechanisms are qualitatively similar. On the other hand, such relations cannot predict the secondary atomization occurring within the nucleate-boiling regime, as the break-up mechanisms within this regime have significantly different characteristics. In this context, the present article recognizes the relevance of the relations devised for ‘cold impacts’, to fit the size of secondary droplets within the film-boiling regime, as the correlation formulated here has a similar form: SMD/D ₀ = f(We, Re) ~ A ₁ We _N ^?0.6 Re ^?0.23 and proposes a new correlation for impacts within the nucleate-boiling regime: SMD/D ₀ = f(We, Re, Ja) ~ A ₂ We _N ^?0.14 Re ^?011 Ja ^?03. These correlations are observed to hold for impacts onto rough surfaces with dimensionless roughness R _a/D ₀ smaller than 2E-3, but not for larger roughness amplitudes, for which the data are quite scattered.     

Control of three-dimensional phase dynamics in a cylinder wake

Recently there has been a surge of new interest in three-dimensional wake patterns. In the present work, we have devised a method to control the spanwise end conditions and wake patterns using “end suction”, which is both continuously-variable and admits transient control. Classical steady-state patterns, such as parallel or oblique shedding or the “chevron” patterns are simply induced. The wake, at a given Reynolds number, is receptive to a continuous range of oblique shedding angles (θ), rather than to discrete angles, and there is excellent agreement with the “cos θ” formula for oblique-shedding frequencies. We show that the laminar shedding regime exists up to Reynolds numbers (Re) of 205, and that the immense disparity among reported critical Re for wake transition (Re = 140–190) can be explained in terms of spanwise end contamination. Our transient experiments have resulted in the discovery of new phenomena such as “phase shocks” and “phase expansions”, which can be explained in terms of a simple model assuming constant normal wavelength of the wake pattern. Peter Monkewitz (Lausanne) also predicts such transient phenomena from a Guinzburg-Landau model for the wake.     

Hydrodynamic characteristics of a rectangular gas-driven inverse liquid-solid fluidized bed

The hydrodynamic characteristics of a rectangular gas-driven inverse liquid-solid fluidized bed (GDFB) using particles of different diameters and densities were investigated in detail. Rising gas bubbles cause a liquid upflow in the riser portion, enabling a liquid downflow that causes an inverse fluidization in the downer portion. Four flow regimes (fixed bed regime, initial fluidization regime, complete fluidization regime, and circulating fluidization regime) and three transition gas velocities (initial fluidization gas velocity, minimum fluidization gas velocity, and circulating fluidization gas velocity) were identified via visual observation and by monitoring the variations in the pressure drop. The axial local bed voidage (ε) of the downer first decreases and then increases with the increase of the gas velocity. Both the liquid circulation velocity and the average particle velocity inside the downer increase with the increase of the gas velocity in the riser, but decrease with the particle loading. An empirical formula was proposed to successfully predict the Richardson-Zaki index “n”, and the predicted ε obtained from this formula has a ±5% relative error when compared with the experimental ε.     

Simultaneous stress and birefringence measurements during uniaxial elongation of polystyrene melts with narrow molecular weight distribution

Tensile stress and flow-induced birefringence have been measured during uniaxial elongation at a constant strain rate of two polystyrene melts with narrow molecular weight distribution. For both melts, the stress- optical rule (SOR) is found to be fulfilled upto a critical stress of 2.7 MPa, independent of strain rate and temperature. Estimation of the Rouse times of the melts, from both the zero-shear viscosity and the dynamic-shear moduli at high frequency, shows that the violation of the SOR occurs when the strain rate multiplied by the Rouse time of the melt exceeds by approximately 3. The presented results indicate that in contrast to current predictions of molecular theories, the regime of extensional thinning observed by Bach et al. (2003) extends well beyond the onset of failure of the SOR, and therefore the onset of chain stretch in the non-Gaussian regime.

Shock waves and equations of state of matter

The physical properties of hot dense matter over a broad domain of the phase diagram are of immediate interest in astrophysics, planetary physics, power engineering, controlled thermonuclear fusion, impulse technologies, enginery, and several special applications. The use of intense shock waves in dynamic physics has made the exotic high-energy density states of matter a subject of laboratory experiments and enabled advancing by many orders of magnitude along the pressure scale to range into the megabars and even gigabars. The present report reviews the contribution of shock-wave methods to the problem of the equation of state (EOS) at extreme conditions. Experimental techniques for high-energy density cumulation, the drivers of intense shock waves, and methods for the fast diagnostics of high-energy matter are considered. It is pointed out that the available high pressure and temperature information covers a broad range of the phase diagram, but only irregularly and, as a rule, is not thermodynamically complete; its generalization can be done only in the form of a thermodynamically complete EOS. As a practical example, construction of multi-phase EOS for iron is presented. The model’s results are shown for numerous shock-wave data, the high-pressure melting and evaporation regions and the critical point of iron.     

Buckling and its effect on the confined flow of a model capsule suspension

The rheology of confined flowing suspensions, such as blood, depends upon the dynamics of the components, which can be particularly rich when they are elastic capsules. Using boundary integral methods, we simulate a two-dimensional model channel through which flows a dense suspension of fluid-filled capsules. A parameter of principal interest is the equilibrium membrane perimeter, parameterized by ξ_o, which ranges from round capsules with ξ_o=1.0 to ξ_o=3.0 capsules with a dog-bone-like equilibrium shape. It is shown that the minimum effective viscosity occurs for ξ_o≈1.6, which forms a biconcave equilibrium shape, similar to a red blood cell. The rheological behavior changes significantly over this range; transitions are linked to specific changes in the capsule dynamics. Most noteworthy is an abrupt change in behavior for ξ_o≈2.0, which correlates with the onset of capsule buckling. The buckled capsules have a more varied orientation and make significant rotational (rotlet) contributions to the capsule–capsule interactions.     

A Variational Model for Dislocations in the Line Tension Limit

We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H ^1/2 type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitiño and Ortiz, but it is also of broader mathematical interest. In the context of the dislocation model we identify the Γ-limit of the energy in all scaling regimes for the number N _? of obstacles. The most interesting regime is N _? ≈|ln ?|/?, where ? is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic, and hence not compact. Thus a key ingredient in the proof is a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results from Alberti, Bouchitté and Seppecher for the two-well problem with a H ^1/2 regularization.     

Simulation of the Brownian coagulation of nanoparticles with initial bimodal size distribution via moment method

The Brownian coagulation of nanoparticles with initial bimodal size distribution, i.e., mode i and j, is numerically studied using the moment method. Evolutions of particle number concentration, geometric average diameter and geometric standard deviation are given in the free molecular regime, the continuum regime, the free molecular regime and transition regime, the free molecular regime and continuum regime, respectively. The results show that, both in the free molecular regime and the continuum regime, the number concentration of mode i and j decreases with increasing time. The evolutions of particle geometric average diameter with different initial size distribution are quite different. Both intra-modal and inter-modal coagulation finally make the polydispersed size distribution become monodispersed. As time goes by, the size distribution with initial bimodal turns to be unimodal and shifts to a larger particle size range. In the free molecular regime and transition regime, the inter-modal coagulation becomes dominant when the number concentrations of mode i and j are of the same order. The effects of the number concentration of mode i and mode j on the evolution of geometric average diameter of mode j are negligible, while the effects of the number concentration of mode j on the evolution of geometric average diameter of mode j is distinct. In the free molecular regime and continuum regime, the higher the initial number concentration of mode j, the more obvious the variation of the number concentration of mode i.     

A double-front structure of detonation wave as the result of phase transitions

Using thermochemical code calculations, we show that the nanographite–nanodiamond phase transition, which may occur in the detonation products of a number of carbon containing explosives, can affect the detonation properties and can cause a specific detonation regime with some unusual peculiarities. Among them, we first note the failure of the Chapman–Jouguet condition and the presence of the sonic plane, where the Mach number is equal to unity, in a detonation product expansion wave at a lower pressure than that at the Chapman–Jouguet point. The peculiarities of this detonation regime are demonstrated by the example of TNT, HNS, and RDX. The computed detonation velocities are in excellent agreement with experiments over a wide range of initial charge densities for all of the investigated explosives. The results of this work allow one to explain, e.g., contradictory experimental data on the detonation pressure and on the length of the reaction zone for TNT. We believe that some other solid–solid, solid–liquid, and liquid–liquid phase transformations in the detonation products may also cause a detonation regime with the same features as shown here for the nanographite–nanodiamond transition. We suggest a computational study that should facilitate proposing detonation experiments strongly arguing in favor of the model presented. PACS 47.40.-x; 47.40.Rs; 64.70.-p; 64.70.Kb; 05.70.-a; 05.70-.CeThis paper was based on the work that was presented at the 19th International Colloquium on the Dynamics of Explosions and Reactive Systems, Hakone, Japan, July 27–August 1, 2003.     

Phenomenological study of parabolic and spherical indentation of elastic-ideally plastic material

A phenomenological study of parabolic and spherical indentation of elastic ideally plastic materials was carried out by using precise results of finite elements calculations. The study shows that no “pseudo-Hertzian” regime occurs during spherical indentation. As soon as the yield stress of the indented material is exceeded, a deviation from the, purely elastic Hertzian contact behaviour is found. Two elastic–plastic regimes and two plastic regimes are observed for materials of very large Young modulus to Yield stress ratio, E/σ_y. The first elastic–plastic regime corresponds to a strong evolution of the indented plastic zone. The first plastic regime corresponds to the commonly called “fully plastic regime”, in which the average indentation pressure is constant and equal to about three times the yield stress of the indented material. In this regime, the contact depth to penetration depth ratio tends toward a constant value, i.e. h_c/h = 1.47. h_c/h is only constant for very low values of yield strain (σ_y/E lower than 5 × 10^?6) when aE¹/Rσ_y is higher than 10,000. The second plastic regime corresponds to a decrease in the average indentation pressure and to a steeper increase in the pile-up. For materials with very large E/σ_y ratio, the second plastic regime appears when the value of the non-dimensional contact radius a/R is lower than 0.01. In the case of spherical and parabolic indentation, results show that the first plastic regime exists only for elastic-ideally plastic materials having an E/σ_y ratio higher than approximately 2.000.