Two-dimensional reactive flow dynamics in cellular detonation waves

This investigation deals with the two-dimensional unsteady detonation characterized by the cellular structure resulting from trajectories of triple-shock configurations formed by the transverse waves and the leading shock front. The time-dependent reactive shock problem considered here is governed by a system of nonlinear hyperbolic conservation laws coupled to a polytropic equation of state and a one-step Arrhenius chemical reaction rate with heat release. The numerical solution obtained allowed us to follow the dynamics of the cellular detonation front involving the triple points, transverse waves and unreacted pockets. The calculations show that the weak tracks observed inside the detonation cells around the points of collision of the triple-shock configurations arise from interactions between the transverse shocks and compression waves generated by the collision. The unreacted pockets of gas formed during the collisions of triple points change form when the activation energy increases. For the self-sustained detonation considered here, the unreacted pockets burn inside the region independent of the downstream rarefaction, and thus the energy released supports the detonation propagation. The length of the region independent of the downstream is approximately the size of one or two detonation cell. Received 13 February 1998 / Accepted 13 August 1998     

Bifurcations in the dynamics of an orthogonal double pendulum

The weakly nonlinear resonant response of an orthogonal double pendulum to planar harmonic motions of the point of suspension is investigated. The two pendulums in the double pendulum are confined to two orthogonal planes. For nearly equal length of the two pendulums, the system exhibits 1:1 internal resonance. The method of averaging is used to derive a set of four first order autonomous differential equations in the amplitude and phase variables. Constant solutions of the amplitude and phase equations are studied as a function of physical parameters of interest using the local bifurcation theory. It is shown that, for excitation restricted in either plane, there may be as many as six pitchfork bifurcation points at which the nonplanar solutions bifurcate from the planar solutions. These nonplanar motions can become unstable by a saddle-node or a Hopf bifurcation, giving rise to a new branch of constant solutions or limit cycle solutions, respectively. The dynamics of the amplitude equations in parameter regions of the Hopf bifurcations is then explored using direct numerical integration. The results indicate a complicated amplitude dynamics including multiple limit cycle solutions, period-doubling route to chaos, and sudden disappearance of chaotic attractors.     

On the influence of low initial pressure and detonation stochastic nature on Mach reflection of gaseous detonation waves

The two-dimensional, time-dependent and reactive Navier–Stokes equations were solved to obtain an insight into Mach reflection of gaseous detonation in a stoichiometric hydrogen-oxygen mixture diluted by 25 % argon. This mixture generates a mode-7 detonation wave under an initial pressure of 8.00 kPa. Chemical kinetics was simulated by an eight-species, forty-eight-reaction mechanism. It was found that a Mach reflection mode always occurs for a planar detonation wave or planar air shock wave sweeping over wedges with apex angles ranging from \(5^\circ \) to \(50^\circ \) . However, for cellular detonation waves, regular reflection always occurs first, which then transforms into Mach reflection. This phenomenon is more evident for detonations ignited under low initial pressure. Low initial pressure may lead to a curved wave front, that determines the reflection mode. The stochastic nature of boundary shape and transition distance, during deflagration-to-detonation transition, leads to relative disorder of detonation cell location and cell shape. Consequently, when a detonation wave hits the wedge apex, there appears a stochastic variation of triple point origin and variation of the angle between the triple point trajectory and the wedge surface. As the wedge apex angle increases, the distance between the triple point trajectory origin and the wedge apex increases, and the angle between the triple point trajectory and the wedge surface decreases exponentially.     

On the reduction of some second-order nonlinear ODEs in physics and mechanics to first-order nonlinear integrodifferential and Abel’s classes of equations

Second-order ordinary differential equations (ODEs) with strong nonlinear stiffness terms (cubic nonlinearities) governing wave motions, dynamic crack propagations, nonlinear oscillations etc. in physics and nonlinear mechanics are analyzed. Selecting as guide line a second-order nonlinear ODE of the form of the forced Duffing equation and using admissible functional transformations it is possible to reduce it to an equivalent first-order nonlinear integrodifferential equation. The reduced equation is exact. In the limits of small or large values of the parameter characterizing this nonlinear problem, it is shown that further reductions lead to a nonlinear ODE of the Abel classes. Taking into account the known exact analytic solutions of this equivalent equation it is proved that there does not exist an exact analytic solution of this type of equations. However, in cases when convenient functional relations connecting all parameters of the corresponding null equation and the characteristics of the driving force exist, approximate analytic solutions to the problem under consideration are provided.     

Asymptotic Analysis of an Elasticity Equation for a Finger-Like Hydraulic Fracture

We derive a novel integral equation relating the fluid pressure in a finger-like hydraulic fracture to the fracture width. By means of an asymptotic analysis in the small height to length ratio limit we are able to establish the action of the integral operator for receiving points that lie within three distinct regions: (1) an outer expansion region in which the dimensionless pressure is shown to be equal to the dimensionless width plus a small correction term that involves the second derivative of the width, which accounts for the nonlocal effects of the integral operator. The leading order term in this expansion is the classic local elasticity equation in the PKN model that is widely used in the oil and gas industry; (2) an inner expansion region close to the fracture tip within which the action of the elastic integral operator is shown to be the same as that of a finite Hilbert transform associated with a state of plane strain. This result will enable pressure singularities and stress intensity factors to be incorporated into analytic models of these finger-like fractures in order to model the effect of material toughness; (3) an intermediate region within which the action of the Fredholm integral operator of the first kind is reduced to a second kind operator in which the integral term appears as a small perturbation which is associated with a convergent Neumann series. These results are important for deriving analytic models of finger-like hydraulic fractures that are consistent with linear elastic fracture mechanics. Submitted to Journal of Elasticity on February 5, 2007. Re-submitted with revisions on May 30, 2007.     

Experimental study of gaseous detonation propagation over acoustically absorbing walls

The paper presents results from an experimental investigation of the propagation of gaseous detonation waves over tube sections lined with acoustically absorbent materials. The measurements were compared with results from control tests in a smooth wall section. The results show the increasing effectiveness of a perforated steel plate, wire mesh and steel wool in attenuating detonation. Received 25 October 2000 / Accepted 20 August 2001     

New analytical approach to energy pumping problem in strongly nonhomogeneous 2dof systems

We present new analytical approach to the problem of energy pumping in strongly non- homogeneous nonlinear two-degree-of-freedom (2DOF) systems with single anchor spring under condition of initial impact. Energy pumping is a passive, almost irreversible transfer of mechanical energy from the main substructure of the system to the light auxiliary attachment. The mechanism of energy pumping in the system under consideration is a resonance capture. The approach is based on application of Laplace transformation to the principal asymptotic approximation of the equations of motion in complex form and using the power expansion of the solution in terms of time. Obtained temporal dependence of the system energetic characteristics gives a tool for estimation of energy pumping efficiency. In particular, we show that the system without an anchor spring in attachment is more efficient than the system with such a spring. Numerical simulations confirm the analytical results.     

An analytical study on the geometrical size effect on phase transitions in a slender compressible hyperelastic cylinder

In this paper, we study phase transitions in a slender circular cylinder composed of a compressible hyperelastic material with a non-convex strain-energy function in a loading process. We aim to construct the asymptotic solutions based on an axisymmetrical three-dimensional setting and use the results to describe the key features observed in the experiments by others. By using a methodology involving coupled series-asymptotic expansions, we derive the normal form equation of the original complicated system of non-linear PDEs. Based on a phase-plane analysis, we manage to deduce the global bifurcation properties and to solve the boundary-value problem analytically. The explicit solutions (including post-bifurcation solutions) in terms of integrals are obtained. The engineering stress-strain curve plotted from the asymptotic solutions can capture the key features of the curve measured in some experiments. Our results can also describe the geometrical size effect as observed in experiments. It appears that the asymptotic solutions obtained shed certain light on the instability phenomena associated with phase transitions in a cylinder.     

Propagation of surface (seismic) waves: Ordinary differential equations with strongly nonlinear damping

Second-order ordinary differential equations (ODEs) with strongly nonlinear damping (cubic nonlinearities) govern surface wave motions that entail nonlinear surface seismic motions. They apply to dynamic crack propagation and nonlinear oscillation problems in physics and nonlinear mechanics. It is shown that the nonlinear surface seismic wave equation (Rayleigh equation) admits several functional transformations and it is possible to reduce it to an equivalent first-order Abel ODE of the second kind in normal form. Based on a recently developed methodology concerning the construction of exact analytic solutions for the type of Abel equations under consideration, exact solutions are obtained for the nonlinear seismic wave (NLSW) equation for initial conditions of the physical problem. The method employed is general and can be applied to a large class of relevant ODEs in mathematical physics and nonlinear mechanics.     

Nonlinear waves in diatomic crystals

A possible improvement of a continuum model for diatomic crystals is examined using continuum limit of a discrete diatomic model. For this purpose, various discrete models of diatomic lattice are compared at the linearized and weakly nonlinear levels. The suitable numbering of the atoms in the lattice is found which is better adopted for continualization than the familiar pair numbering introducing two sub-lattices. The coupled governing partial nonlinear differential equations for longitudinal strain and relative distance between the atoms are obtained in the continuum limit that allows us to describe localization of the strains due to the presence of the atoms of two kinds. It is found, that the equations obtained possess two kinds of localized wave solutions, one related to the acoustical branch and the other one related to the optical branch.     

Orbital stability of periodic waves for the nonlinear Schrödinger equation

The nonlinear Schr?dinger equation has several families of quasi-periodic traveling waves, each of which can be parametrized up to symmetries by two real numbers: the period of the modulus of the wave profile, and the variation of its phase over a period (Floquet exponent). In the defocusing case, we show that these travelling waves are orbitally stable within the class of solutions having the same period and the same Floquet exponent. This generalizes a previous work (Gallay and Haragus, J. Diff. Equations, 2007) where only small amplitude solutions were considered. A similar result is obtained in the focusing case, under a non-degeneracy condition which can be checked numerically. The proof relies on the general approach to orbital stability as developed by Grillakis, Shatah, and Strauss, and requires a detailed analysis of the Hamiltonian system satisfied by the wave profile.     

Transition waves in bistable structures. I. Delocalization of damage

We consider chains of dimensionless masses connected by breakable bistable links. A non-monotonic piecewise linear constitutive relation for each link consists of two stable branches separated by a gap of zero resistance. Mechanically, this model can be envisioned as a ”twin-element” structure which consists of two links (rods or strands) of different lengths joined by the ends. The longer link does not resist to the loading until the shorter link breaks. We call this construction the waiting link structure. We show that the chain of such strongly non-linear elements has an increased in-the-large stability under extension in comparison with a conventional chain, and can absorb a large amount of energy. This is achieved by two reasons. One is an increase of dissipation in the form of high-frequency waves transferring the mechanical energy to heat; this is a manifestation of the inner instabilities of the bonds. The other is delocalization of the damage of the chain. The increased stability is a consequence of the distribution of a partial damage over a large volume of the body instead of its localization, as in the case of a single neck formation in a conventional chain. We optimize parameters of the structure in order to improve its resistance to a slow loading and show that it can be increased significantly by delocalizing a damage process. In particular, we show that the dissipation is a function of the gap between the stable branches and find an optimal gap corresponding to maximum energy consumption under quasi-static extension. The results of numerical simulations of the dynamic behavior of bistable chains show that these chains can withstand without breaking the force which is several times larger than the force sustained by a conventional chain. The formulation and results are also related to the modelling of compressive destruction of a porous material or a frame construction which can be described by a two-branched diagram with a large gap between the branches. We also consider an extension of the model to multi-link chain that could imitate plastic behavior of material.     

Nonlinear waves in fluid flow through a viscoelastic tube

A system of nonlinear equations for describing the perturbations of the pressure and radius in fluid flow through a viscoelastic tube is derived. A differential relation between the pressure and the radius of a viscoelastic tube through which fluid flows is obtained. Nonlinear evolutionary equations for describing perturbations of the pressure and radius in fluid flow are derived. It is shown that the Burgers equation, the Korteweg-de Vries equation, and the nonlinear fourth-order evolutionary equation can be used for describing the pressure pulses on various scales. Exact solutions of the equations obtained are discussed. The numerical solutions described by the Burgers equation and the nonlinear fourth-order evolutionary equation are compared.     

Investigation of the detonation regimes in gaseous mixtures with suspended starch particles

The existence of a secondary discontinuity at the rear of a detonation front shown in experiments by Peraldi and Veyssiere (1986) in stoichiometric hydrogen-oxygen mixtures with suspended 20-m starch particles has not been explained satisfactorily. Recently Veyssiere et al. (1997) analyzed these results using a one-dimensional (1-D) numerical model, and concluded that the heat release rate provided by the burning of starch particles in gaseous detonation products is too weak to support a double-front detonation (DFD), in contrast to the case of hybrid mixtures of hydrogen-air with suspended aluminium particles in which a double-front detonation structure was observed by Veyssiere (1986). A two-dimensional (2-D) numerical model was used in the present work to investigate abovementioned experimental results for hybrid mixtures with starch particles. The formation and propagation of the detonation has been examined in the geometry similar to the experimental tube of Peraldi and Veyssiere (1986), which has an area change after 2 m of propagation from the ignition point from a 69 mm dia. section to a 53 mm 53 mm square cross section corresponding to a 33% area contraction. It is shown that the detonation propagation regime in these experiments has a different nature from the double-front detonation observed in hybrid mixtures with aluminium particles. The detonation propagates as a pseudo-gas detonation (PGD) because starch particles release their heat downstream of the CJ plane giving rise to a non-stationary compression wave. The discontinuity wave at the rear of the detonation front is due to the interaction of the leading detonation front with the tube contraction, and is detected at the farthest pressure gauge location because the tube length is insufficient for the perturbation generated by the tube contraction to decay. Thus, numerical simulations explain experimental observations made by Peraldi and Veyssiere (1986). Received 5 July 1997 / Accepted 13 July 1998     

The singularity in weakly nonlinear fracture mechanics

Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack's tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale ℓ from a crack's tip, significant logr displacements and 1/r displacement-gradient contributions arise. Whereas in LEFM the 1/r singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is also necessary in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As ℓ is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics.     

A NUMERICAL AND ANALYTICAL STUDY ON A TAIL-FLAPPING MODEL FOR FISH FAST C-START＊

The force production physics and the flow control mechanism of fish fast C-start are studied numerically and theoretically by using a tail-flapping model. The problem is simplified to a 2-D foil that rotates rapidly to and fro on one side about its fixed leading edge in water medium. The study involves the simulation of the flow by solving the two-dimensional unsteady incompressible Navier-Stokes equations and employing a theoretical analytic modeling approach. Firstly, reasonable thrust magnitude and its time history are obtained and checked by fitting predicted results coming from these two approaches. Next, the flow fields and vortex structures are given, and the propulsive mechanism is interpreted. The results show that the induction of vortex distributions near the trailing edge of the tail are important in the time-averaged thrust generation, though the added inertial effect plays an important role in producing an instant large thrust especially in the first stage. Furthermore, dynamic and energetic effects of some kinematic controlling factors are discussed. For enhancing the time-averaged thrust but keeping a favorable ratio of it to time-averaged input power within the limitations of muscle ability, it is recommended to have a larger deflection amplitude in a limited time interval and with no time delay between the to-and-fro strokes. The project supported by the CAS (KJCX-SW-L04)     

A numerical study of two- and three-dimensional detonation dynamics of pulse detonation engine by the CE／SE method

In this paper, the CE/SE method is developed to simulate the two- and three-dimensional flow-field of Pulse Detonation Engine (PDE). The conservation equations with stiff source terms for chemical reaction are solved in two steps. The detailed analysis of computational results of a PDE with a single detonation tube and a PDE with five detonation tubes are given in this paper. Complex wave systems are observed inside and outside a PDE. For a PDE with 5 detonation tubes, there is a big bow shock produced from a number of little shocks near the open ends of tubes. A lot of vortexes interact with shocks and a large expansion wave propagates forward and backward with respect to the PDE in a semi-oval shape.The project supported by the National Natural Science Foundation of China (59906005), the Teaching and Research Award Program for Outstanding Young Teachers in High Education Institutions of MOE, China     

Experimental study and kinetic modeling of the thermal decomposition of gaseous monomethylhydrazine. Application to detonation sensitivity

The thermal decomposition of gaseous monomethylhydrazine has been studied in a 38.4 mm i.d. shock tube behind a reflected shock wave at 1040–1370 K, 140–455 kPa and in mixtures containing 97 to 99 mol% argon, by using MMH absorption at 220 nm. A chemical kinetic model based on MMH decomposition profiles has been developed. This model has been used, with some assumptions, to evaluate the detonation sensitivity of pure gaseous MMH. This compound is found to be much less sensitive to detonation than hydrazine. An abridged version of this paper was presented at the 15th Int. Colloquium on the Dynamics of Explosions and Reactive Systems at Boulder, Colorado, from July 30 to August 4, 1995