Reactive Euler Equations Research Articles

A numerical study of 2D detonation waves with adaptive finite volume methods on unstructured grids

In this paper, a framework of adaptive finite volume solutions for the reactive Euler equations on unstructured grids is proposed. The main ingredients of the algorithm include a second order total variation diminishing Runge–Kutta method for temporal discretization, and the finite volume method with piecewise linear solution reconstruction of the conservative variables for the spatial discretization in which the least square method is employed for the reconstruction, and weighted essentially nonoscillatory strategy is used to restrain the potential numerical oscillation. To resolve the high demanding on the computational resources due to the stiffness of the system caused by the reaction term and the shock structure in the solutions, the h-adaptive method is introduced. OpenMP parallelization of the algorithm is also adopted to further improve the efficiency of the implementation. Several one and two dimensional benchmark tests on the ZND model are studied in detail, and numerical results successfully show the effectiveness of the proposed method.

Initiation characteristics of wedge-induced oblique detonation waves in a stoichiometric hydrogen-air mixture

The initiation features of two-dimensional, oblique detonations from a wedge in a stoichiometric hydrogen-air mixture are investigated via numerical simulations using the reactive Euler equations with detailed chemistry. A parametric study is performed to analyze the effect of inflow pressure P0, and Mach number M0 on the initiation structure and length. The present numerical results demonstrate that the two transition patterns, i.e., an abrupt transition from a multi-wave point connecting the oblique shock and the detonation surface and a smooth transition via a curved shock, depend strongly on the inflow Mach number, while the inflow pressure is found to have little effect on the oblique shock-to-detonation transition type. The present results also reveal a slightly more complex structure of abrupt transition type in the case of M0=7.0, consisting of various chemical and gasdynamic processes in the shocked gas mixtures. The present results show quantitatively that the initiation length decreases with increasing M0, primarily due to the increase of post-shock temperature. Furthermore, the effect of M0 on initiation length is independent of P0, but given the same M0, the initiation length is found to be inversely proportional to P0. Theoretical analysis based on the constant volume combustion (CVC) theory is also performed, and the results are close to the numerical simulations in the case of high M0 regardless of P0, demonstrating that the post-oblique-shock condition, i.e., post-shock temperature, is the key parameter affecting the initiation. At decreasing M0, the CVC theory breaks down, suggesting a switch from chemical kinetics-controlled to a wave-controlled gasdynamic process. For high inflow pressure P0 at decreasing M0, the CVC theoretical estimations depart from numerical results faster than those of low P0, due to the presence of the non-monotonic effects of chemical kinetic limits in hydrogen oxidation at high pressure.

Direct detonation initiation with thermal deposition due to pore collapse in energetic materials – towards the coupling between micro- and macroscale

In this work we investigate the initiation of detonations in energetic materials through thermal power deposition due to pore collapse. We solve the reactive Euler equations, with the energy equation augmented by a power deposition term. The deposition term is partially based on previous results of simulations of pore collapse at the microscale, modelled at the macroscale as hotspots. It is found that a critical size of the hotspots exists. If the hotspots exceed the critical size, direct initiation of detonation can be achieved even with a low power input, in contrast to the common assumption that a sufficient power is necessary to initiate detonation. We show that sufficient power is necessary only when the size of the hotspots is below the critical size. In this scenario, the so-called ‘explosion in the explosion’, the initial ignition does not lead to a detonation directly, but detonation occurs later as a result of shock-to-detonation transition in the region processed by the shock wave generated by the initial ignition.

Numerical study of hollow rotating detonation engine with different fuel injection area ratios

Three-dimensional numerical simulations are performed to study rotating detonation engines (RDEs) with a hollow combustor. The computations are based on the reactive Euler equations. The reactions of stoichiometric hydrogen–air mixtures are calculated by the one-step Arrhenius chemistry model. The hollow RDE combustors with different fuel injection area ratios on the head end wall are considered. The fuel injection area ratios range from 55.6% to 88.9%. The number of rotating detonation waves in the flow field of the hollow combustor is found to be related to the fuel injection area ratio. In the hollow combustors with fixed dimensions, the number of detonation waves increases when the fuel injection area ratio is higher. In addition, a comparative analysis of the hollow RDE model and the conventional annular RDE model is performed with respect to the flow field and fuel injection conditions. The propulsive performance of the hollow RDE with different fuel injection area ratios is also evaluated and compared with experimental data and other simulation results. It is demonstrated from several aspects that the removal of the inner wall does not sacrifice the propulsive performance of the RDE.

Numerical simulations of cellular detonation diffraction in a stable gaseous mixture

In this paper, the diffraction phenomenon of gaseous cellular detonations emerging from a confined tube into a sudden open space is simulated using the reactive Euler equations with a two-step Arrhenius chemistry model. Both two-dimensional and axisymmetric configurations are used for modeling cylindrical and spherical expansions, respectively. The chemical parameters are chosen for a stable gaseous explosive mixture in which the cellular detonation structure is highly regular. Adaptive mesh refinement (AMR) is used to resolve the detonation wave structure and its evolution during the transmission. The numerical results show that the critical channel width and critical diameter over the detonation cell size are about 13±1 and 25±1, respectively. These numerical findings are comparable with the experimental observation and confirm again that the critical channel width and critical diameter differ essentially by a factor close to 2, equal to the geometrical scaling based on front curvature theory. Unlike unstable mixtures where instabilities manifested in the detonation front structure play a significant role during the transmission, the present numerical results and the observed geometrical scaling provide again evidence that the failure of detonation diffraction in stable mixtures with a regular detonation cellular pattern is dominantly caused by the global curvature due to the wave divergence resulting in the global decoupling of the reaction zone with the expanding shock front.

Evolution of detonation formation initiated by a spatially distributed, transient energy source

Detonations usually form through either direct initiation or deflagration-to-detonation transition (DDT). In this work, a detonation initiation process is introduced that shows attributes from each of these two processes. Energy is deposited into a finite volume of fluid in an amount of time that is similar to the acoustic time scale of the heated fluid volume. Two-dimensional simulations of the reactive Euler equations are used to solve for the evolving detonation initiation process. The results show behaviour similar to both direct initiation and DDT. Localized reaction transients are shown to be intimately related to the appearance of a detonation. Thermomechanical concepts are used to provide physical interpretations of the computational results in terms of the interaction between compressibility phenomena on the acoustic time scale and localized, spatially resolved, chemical energy addition on a heat-addition time scale.

Three-dimensional detonation simulations with the mapped WENO-Z finite difference scheme

Abstract We perform a very long time numerical simulation to capture the three-dimensional detonation structures in a rectangular duct by solving the reactive Euler equations using the high order/resolution WENO-Z conservative finite difference scheme (Gao et al. J. Sci. Comput. 55 : 351–371, 2013). In the algorithm, the perfectly matched layer (PML) absorbing boundary condition (ABC) for the reactive Euler equations is used to reduce the spurious wave reflection from the open left boundary which allows one to use a significant smaller truncated physical domain. Moreover, a tangent grid mapping is used to enhance the grid resolution within the half reaction zone that greatly reduces the memory usage and computational time compared with solving the problem with a uniform grid. The initial Zeldovich-von Neumann-Doring (ZND) profile of two classical stable and slightly unstable detonation waves is perturbed to generate the rectangular in-phase, diagonal in-phase and spinning detonation structures. Depending on the initial perturbation, the stable case shows the presence of a rectangular mode and a diagonal mode of the detonation front, which are suggested to be geometrically similar. The slightly unstable case, as expected, generates the spinning detonations instead in a narrow duct. The results show that a short time simulation is insufficient to capture the cellular detonation structures. The width-to-length ratio of the cellular patterns depends on the gas properties only, but independent of the perturbation of the initial conditions.

Adaptive simulations of cavity-based detonation in supersonic hydrogen–oxygen mixture

Two-dimensional reactive Euler equations with a detailed reaction model where the molar ratio of the combustible mixture H2/O2/Ar is 2:1:7 under the condition of pressure 6.67 kPa and temperature 298 K, are solved numerically with adaptive mesh refinement method to investigate detonation combustion using a hot jet initiation in cavity-based channels filled with the supersonic combustible mixture. Results show that from the comparison between the simulations in a cavity-based channel and a straight channel without any cavity embedded, it is indicated that the cavity can help realize detonation initiation in the combustible mixture with a hot jet. It is suggested that detonation initiation can be realized using a relatively weaker hot jet in cavity-based channels filled with the supersonic combustible mixture compared with that in straight channels without a cavity embedded. The cavity also plays a significant role in detonation propagation in the supersonic combustible mixture. After the hot jet is shut down, the acoustic wave generated by the subsonic combustion in the cavity can accelerate detonation propagation through a subsonic channel and result in the formation of a slightly overdriven detonation eventually. For a given flow with a shadow cavity embedded, there should exist a minimum cavity width Lmin. When the width is below Lmin, only some pressure oscillations in the cavity can make some impacts on detonation initiation and propagation. Otherwise, cavity oscillations can be generated which can greatly accelerate detonation initiation and propagation in the supersonic combustible mixture. For the shadow cavity, purely increasing the cavity depth does not have any more influence on detonation combustion. However, if the cavity is a deep one, it can play an important role in accelerating detonation initiation and propagation in the supersonic combustible mixture due to resonant oscillations.

The Zeldovich spontaneous reaction wave propagation concept in the fast/modest heating limits

Quantitative mathematical models describe planar, spontaneous, reaction wave propagation (Zeldovich, Combust. Flame, vol. 39, 1980, pp. 211–214) in a finite hot spot volume of reactive gas. The results describe the complete thermomechanical response of the gas to a one-step, high-activation-energy exothermic reaction initiated by a tiny initial temperature non-uniformity in a gas at rest with uniform pressure. Initially, the complete conservation equations, including all transport terms, are non-dimensionalized to identify parameters that quantify the impact of viscosity, conduction and diffusion. The results demonstrate unequivocally that transport terms are tiny relative to all other terms in the equations, given the relevant time and length scales. The asymptotic analyses, based on the reactive Euler equations, describe both induction and post-induction period models for a fast heat release rate (induction time scale short compared to the acoustic time of the spot), as well as a modest heat release rate (induction time scale equivalent to the acoustic time). Analytical results are obtained for the fast heating rate problem and emphasize the physics of near constant-volume heating during the induction period. Weak hot spot expansion is the source of fluid expelled from the original finite volume and is a ‘piston-effect’ source of acoustic mechanical disturbances beyond the spot. The post-induction period is characterized by the explosive appearance of an ephemeral, spatially uniform high-temperature, high-pressure spot embedded in a cold, low-pressure environment. In analogy with a shock tube the subsequent expansion process occurs on the acoustic time scale of the spot and will be the source of shocks propagating beyond the spot. The modest heating rate induction period is characterized by weakly compressible phenomena that can be described by a novel system of linear wave equations for the temperature, pressure and induced velocity perturbations driven by nonlinear chemical heating, which provides physical insights difficult to obtain from the more familiar ‘Clarke equation’. When the heating rate is modest, reaction terms in the post-induction period Euler equations exhibit a form of singular behaviour in the high-activation-energy limit, implying the need to use a nonlinear exponential scaling for time and space, developed originally to describe spatially uniform thermal explosions (Kassoy, Q. J. Mech. Appl. Maths, vol. 30, 1977, pp. 71–89). Here again the result will be the explosive appearance of an ephemeral spatially uniform high-temperature, high-pressure hot spot. These results demonstrate that an initially weak temperature non-uniformity in a finite hot spot can be the source of acoustic and shock wave mechanical disturbances in the gas beyond the spot that may be related to rocket engine instability and engine knock.

Numerical investigations on reignition behavior of detonation diffraction

In this paper, by adopting a fifth-order weighted essentially non-oscillatory (WENO) scheme with a third-order TVD Runge–Kutta time stepping method for two-dimensional reactive Euler equations, a parallel code is developed, and reignition behavior after a self-sustaining detonation from the tube into free space filled with H2/O2 mixtures is investigated. The numerical results show that the initial pressure has a great influence on the detonation cellular width, and that as the initial pressure increases, the cellular width gradually decreases and the cellular shape changes from irregular structure to regular structure, demonstrating the detonation instability to stability transition. When the initial pressure is larger than 1.2 atm, the detonation wave expands over the edge of the splitter plate, reignition can come into being because enough transverse waves collide with each other at the leading edge of the expanding front. When the initial pressure is 1.2 atm, hot spots appear on the front, and ignite the combustible gas near the hot spots after detonation diffraction. When the initial pressure is 1.0 atm, reignition fails. These findings hint that a critical initial pressure exists between 1.0–1.2 atm for direct reignition after detonation diffraction.

Adaptive mesh refinement based simulations of three-dimensional detonation combustion in supersonic combustible mixtures with a detailed reaction model

Detonation combustion initiated with a hot jet in supersonic H2-O2-Ar mixtures is investigated by large-scale three-dimensional (3D) simulations conducted in Tianhe-2 computing system with adaptive mesh refinement method. The reactive Euler equations are utilized as governing equations with a detailed reaction model where the molar ratio of the combustible mixture is 2:1:7 under the condition of pressure 10 kPa and temperature 298 K. Results show that the Mach stem surface which is formed after the shock surface reflection on the upper wall is actually a local overdriven detonation. The side walls in 3D simulations can play an important role in detonation initiation in supersonic combustible mixtures, because they can help realize triple lines collisions and reflections during the initiation process. The width of the channel has an important influence on the strength of side-wall reflections, and under certain condition there might exist a critical width between the front and back sides of the channel for the successful initiation. Both the two-dimensional (2D) and 3D detonations are overdriven and have constant but different overdrive degrees after their complete initiations. Although the overdrive degree of the 3D detonation is smaller than that of the 2D case, more complex and irregular detonation fronts can be observed in the 3D case compared with the 2D detonation, which is likely because of the propagation of transverse waves and collisions of triple lines in multi-directions in 3D detonations. When the hot jet is shut down, the newly formed 2D Chapman-Jouguet (CJ) detonation has almost the same characteristic parameters with the corresponding 3D case, indicating that the 2D instabilities can be perfectly preserved in 3D simulations. However, the slapping wave reflections on the side walls in the 3D detonation result in the second oscillation along with the main one, which presents stronger instabilities compared with the 2D case. The inherent stronger 3D instabilities are also verified through the quantitative comparison between the 2D and 3D cases where the 3D result always shows stronger fluctuations than the 2D case.

Set-valued solutions for non-ideal detonation

The existence and structure of a steady-state gaseous detonation propagating in a packed bed of solid inert particles are analyzed in the one-dimensional approximation by taking into consideration frictional and heat losses between the gas and the particles. A new formulation of the governing equations is introduced that eliminates the difficulties with numerical integration across the sonic singularity in the reactive Euler equations. With the new algorithm, we find that when the sonic point disappears from the flow, there exists a one-parameter family of solutions parameterized by either pressure or temperature at the end of the reaction zone. These solutions (termed “set-valued” here) correspond to a continuous spectrum of the eigenvalue problem that determines the detonation velocity as a function of a loss factor.

Theory of weakly nonlinear self-sustained detonations

We propose a theory of weakly nonlinear multidimensional self-sustained detonations based on asymptotic analysis of the reactive compressible Navier–Stokes equations. We show that these equations can be reduced to a model consisting of a forced unsteady small-disturbance transonic equation and a rate equation for the heat release. In one spatial dimension, the model simplifies to a forced Burgers equation. Through analysis, numerical calculations and comparison with the reactive Euler equations, the model is demonstrated to capture such essential dynamical characteristics of detonations as the steady-state structure, the linear stability spectrum, the period-doubling sequence of bifurcations and chaos in one-dimensional detonations and cellular structures in multidimensional detonations.

High Order Numerical Simulation of Detonation Wave Propagation Through Complex Obstacles with the Inverse Lax-Wendroff Treatment

AbstractThe high order inverse Lax-Wendroff (ILW) procedure is extended to boundary treatment involving complex geometries on a Cartesian mesh. Our method ensures that the numerical resolution at the vicinity of the boundary and the inner domain keeps the fifth order accuracy for the system of the reactive Euler equations with the two-step reaction model. Shock wave propagation in a tube with an array of rectangular grooves is first numerically simulated by combining a fifth order weighted essentially non-oscillatory (WENO) scheme and the ILW boundary treatment. Compared with the experimental results, the ILW treatment accurately captures the evolution of shock wave during the interactions of the shock waves with the complex obstacles. Excellent agreement between our numerical results and the experimental ones further demonstrates the reliability and accuracy of the ILW treatment. Compared with the immersed boundary method (IBM), it is clear that the influence on pressure peaks in the reflected zone is obviously bigger than that in the diffracted zone. Furthermore, we also simulate the propagation process of detonation wave in a tube with three different widths of wall-mounted rectangular obstacles located on the lower wall. It is shown that the shock pressure along a horizontal line near the rectangular obstacles gradually decreases, and the detonation cellular size become large and irregular with the decrease of the obstacle width.

Numerical study of oblique detonation wave initiation in a stoichiometric hydrogen-air mixture

Two-dimensional, oblique detonations induced by a wedge are simulated using the reactive Euler equations with a detailed chemical reaction model. The focus of this study is on the oblique shock-to-detonation transition in a stoichiometric hydrogen-air mixture. A combustible, gas mixture at low pressure and high temperature, corresponding to the realistic, inflow conditions applied in oblique detonation wave engines, is presented in this study. At practical flight conditions, the present numerical results illustrate that oblique detonation initiation is achieved through a smooth transition from a curved shock, which differs from the abrupt transition depicted in the previous studies. The formation mechanism of this smooth transition is discussed and a quantitative analysis is carried out by defining a characteristic length for the initiation process. The dependence of the initiation length on different parameters including the wedge angle, flight Mach number, and inflow Mach number is discussed. Despite the hypothetical nature of the simulation configuration, the present numerical study uses parameters we deem relevant to practical conditions and provides important observations for which future investigations can benefit from in reaching toward a rigorous theory of the formation and self-sustenance of oblique detonation waves.

Adaptive Multiresolution Computations Applied to Detonations

Abstract A space-time adaptive method is presented for the reactive Euler equations describing chemically reacting gas flow where a two species model is used for the chemistry. The governing equations are discretized with a finite volume method and dynamic space adaptivity is introduced using multiresolution analysis. A time splitting method of Strang is applied to be able to consider stiff problems while keeping the method explicit. For time adaptivity an improved Runge–Kutta–Fehlberg scheme is used. Applications deal with detonation problems in one and two space dimensions. A comparison of the adaptive scheme with reference computations on a regular grid allows to assess the accuracy and the computational efficiency, in terms of CPU time and memory requirements.

A short note on the counter-intuitive spurious behaviors in stiff reacting flow

A well known spurious numerical phenomenon may occur in solving stiff detonation problems due to the under-resolved numerical solution in both space and time. Most people believe that decreasing numerical dissipation or stiffness will delay or eliminate the onset of spurious numerical phenomenon. However, several counter-intuitive spurious behaviors were observed by H.C. Yee et al. (2013) [10] recently and the mechanism of the generation of these strange phenomena remains an open question. The goal of this short note is to give a reasonable explanation for these counter-intuitive spurious behaviors existing in the detonation problems (the simplified 2×2 system and the reactive Euler equations) with stiff reacting source terms and discontinuities. In developing the mechanism of spurious numerical phenomenon in detonation problems, we find the parameters of the intermediate state are very important because they determine whether the spurious phenomenon will happen or not. Furthermore, these counter-intuitive spurious behaviors are mainly due to the oscillation of those intermediate state parameters as the time step or grid is refined gradually. These findings may help us to get a further understanding of some of the difficulties in numerical combustion and problems with stiff nonlinear source terms and discontinuities in general.

Mean structure of one-dimensional unstable detonations with friction

AbstractThis investigation deals with the study of the mean structure of a mildly unstable non-ideal detonation wave. The analysis is based on the integration of one-dimensional reactive Euler equations with friction forces using a third-order Runge–Kutta scheme and a fifth-order weighted essentially non-oscillatory (WENO5) spatial discretization. A one-step Arrhenius reaction mechanism is used for modelling the chemical reaction. When the frictional forces are active, the limit cycle based on the post-shock pressure reveals an enhanced pulsating behaviour of the downstream subsonic reaction zone compared to the ideal case. The results show that the detonation-velocity deficit increases as the mean reaction zone becomes thicker compared to the generalized ZND model. A new master equation, based on the Favre-averaged quantities, is derived and analysed along with new sonicity and thermicity conditions. The analysis of the species, momentum and energy balances reveals that the presence of mechanical fluctuations within the reaction zone constitutes another source of energy withdrawal, meaning that the detonation deviates from its laminar structure. Furthermore, the compressibility of the flow is analysed and the relationships between the fluctuations of temperature, velocity and reactive scalar are discussed in terms of strong Reynolds analogies.

Flame acceleration and transition from deflagration to detonation in hydrogen explosions

Computational Fluid Dynamics solvers are developed for explosion modelling and hazards analysis in Hydrogen air mixtures. The work is presented in two parts. These include firstly a numerical approach to simulate flame acceleration and deflagration to detonation transition (DDT) in hydrogen–air mixture and the second part presents comparisons between two approaches to detonation modelling. The detonation models are coded and the predictions in identical scenarios are compared. The DDT model which is presented here solves fully compressible, multidimensional, transient, reactive Navier–Stokes equations with a chemical reaction mechanism for different stages of flame propagation and acceleration from a laminar flame to a highly turbulent flame and subsequent transition from deflagration to detonation. The model has been used to simulate flame acceleration (FA) and DDT in a 2-D symmetric rectangular channel with 0.04 m height and 1 m length which is filled with obstacles. Comparison has been made between the predictions using a 21-step detailed chemistry as well as a single step reaction mechanism. The effect of initial temperature on the run-up distances to DDT has also been investigated.In the second part, one detonation solver is developed based on the solution of the reactive Euler equations while the other solver has a simpler approach based on Chapman–Jouguet model and the programmed CJ burn method. Comparison has shown that the relatively simple CJ burn approach is unable to capture some very important features of detonation when there are obstacles present in the cloud.

Multi-temperature fluid-dynamic model equations from kinetic theory in a reactive gas: The steady shock problem

Starting from a simple kinetic model for a chemical reaction, multi-temperature reactive Euler equations are derived for physical regimes in which evolution is driven by elastic collisions within the same species and mechanical relaxation is faster than the thermal one. The achieved hydrodynamic equations, where all inhomogeneous exchange rates take analytical closed form for simple collision models, are then used for the analysis of the steady shock problem. Results indicate that smooth shock profiles occurring for slightly supersonic flows bifurcate to weak solutions (jump discontinuity followed by a smooth tail) for increasing Mach number.