Reactive Navier Stokes Equations Research Articles

Cellular combustion at the transition of a spherical flame front to a flat front at the initiated ignition of methane–air, methane–oxygen and n-pentane–air mixtures

High-speed cinematography was used to study the transition of a spherical flame front to a flat front in n -pentane–air and methane–air mixtures initiated by a spark discharge. Cellular flame structures were observed at the transition. Modeling based on compressible reactive Navier–Stokes equations at a low Mach number showed a qualitative agreement with experimental data.

Hydrogen–oxygen flame acceleration and deflagration-to-detonation transition in three-dimensional rectangular channels with no-slip walls

Hydrogen–oxygen flame acceleration and the transition from deflagration to detonation (DDT) in channels with no-slip walls are studied using high resolution simulations of 3D reactive Navier–Stokes equations, including the effects of viscosity, thermal conduction, molecular diffusion, real equation of state and detailed (reduced) chemical reaction mechanism. The acceleration of the flame propagating from the closed end of a channel, which is a key factor for understanding of the mechanism of DDT, is thoroughly studied. The three dimensional modeling of the flame acceleration and DDT in a semi-closed rectangular channel with cross section 10 × 10 mm and length 250 mm confirms validity of the mechanism of deflagration-to-detonation transition, which was proposed earlier theoretically and verified using 2D simulations. We show that 3D model contrary to 2D models allows to understand clearly the meaning of schlieren photos obtained in experimental studies. The “numerical schlieren” and “numerical shadowgraph” obtained using 3D calculations clarify the meaning of the experimental schlieren and shadow photos and some earlier misinterpretations of experimental data.

Three-Dimensional Numerical Simulations of Spherical Flame Evolutions in Shock and Reshock Accelerated Flows

The three-dimensional spherical flame evolutions induced by incident and reflected shock waves in a rectangular shock tube are numerically simulated using the compressible reactive Navier–Stokes equations with a single-step Arrhenius chemical reaction. Four cases, including variable parameters of flame size and number, shock wave strength, and mixture reactivity, are considered in order to investigate the effects of these parameters on the flame evolutions and detonation onsets. The three-dimensional visualized results and the time-dependent integral and statistical results for the shock–flame interactions are obtained. The morphology of the flame evolutions for all cases studied shows the severe distortion, expansion, and corrugation of flame disturbed by shock waves, especially by a reflected shock wave. The unstable flame produces the three-dimensional reactive shock bifurcation (RSB) structure, as the flame number, the shock wave strength, or the mixture reactivity increase. Further, detonations can occur in a later stage of flame evolution at the different three-dimensional spatial locations for different cases through the shock–detonation–transition (SDT) mechanism. Unlike the three-dimensional spatial dissimilarities of flame patterns and detonation initiation locations among cases studied, the time-dependent integral and statistical properties of flame developments prior to detonation onset show the more similar behaviors for all cases. During the passage of shock and reshock waves, the spherical flames are compressed and distorted, resulting in the vorticity deposition within flame and the well-mixing between the burned and unburned gases. Physical process dominates the compression phases of the flame evolution. Following the passage of the shock and reshock waves, the well-mixing facilitates the chemical heat release and expands (accelerates) the distorted flame. The chemical process becomes more prominent, especially under the reshock condition. In the flame expansion phases, the baroclinic effect is weakened, and the vortex stretching effect is enhanced with the development of flame. In addition, the well-mixing by the passages of shock waves promotes the chemical reaction of the flame, which in turn burns out the mixing zone of flame and therefore inhibits the mixing.

Stability of viscous detonations for Majda’s model

Using analytical and numerical Evans-function techniques, we examine the spectral stability of strong-detonation-wave solutions of a version of Majda’s scalar model for a reacting gas mixture with an Arrhenius-type ignition function. We introduce an energy estimate to limit possible unstable eigenvalues to a compact region in the unstable complex half plane, and we use a numerical approximation of the Evans function to search for possible unstable eigenvalues in this region. Our results show, for the parameter values tested, that these waves are spectrally stable. Combining these numerical results with the pointwise Green function analysis of Lyng, Raoofi, Texier, & Zumbrun [G. Lyng, M. Raoofi, B. Texier, K. Zumbrun, Pointwise Green function bounds and stability of combustion waves, J. Differential Equations 233 (2) (2007) 654–698.], we conclude that these waves are nonlinearly stable. This represents the first demonstration of nonlinear stability for detonation-wave solutions of a Majda-type model without a smallness assumption. Notably, our results indicate that, for this simplified, scalar model, there does not occur, either in a normal parameter range or in the limit of high activation energy, Hopf bifurcation to “galloping” or “pulsating” solutions as is observed in the full reactive Navier–Stokes equations. This answers in the negative, for this model, a question posed by Majda as to whether such scalar detonation models capture this aspect of detonation behavior.

Effect of chemical reactivity on the detonation initiation in shock accelerated flow in a confined space

The interactions of a spherical flame with an incident shock wave and its reflected shock wave in a confined space were investigated using the three-dimensional reactive Navier-Stokes equations, with emphasis placed on the effect of chemical reactivity of mixture on the flame distortion and detonation initiation after the passage of the reflected shock wave. It is shown that the spatio-temporal characteristics of detonation initiation depend highly on the chemical reactivity of the mixture. When the chemical reactivity enhances, the flame can be severely distorted to form a reactive shock bifurcation structure with detonations initiating at different three-dimensional spatial locations. Moreover, the detonation initiation would occur earlier in a mixture of more enhanced reactivity. The results reveal that the detonations arise from hot spots in the unburned region which are initiated by the shock-detonation-transition mechanism.

Direct numerical simulations of high speed reactive mixing layers

The present work is devoted to the analysis of ignition phenomena in high speed turbulent reactive flows. From the computational modelling point of view, this requires to account accurately for the competition between molecular diffusion effects and chemical kinetics as well as complex flowfield structures that may feature shock and expansion waves. Some results of direct numerical simulations of the multicomponent, compressible, reactive Navier-Stokes equations are reported here for the particular case of a two-dimensional mixing layer. Special emphasis is placed on the effects associated with heat release.

Numerical study on flame acceleration and DDT in an inclined array of cylinders using an AMR technique

Flame acceleration and deflagration-to-detonation transition in an inclined array of cylinders embedded in a H2–air gas are studied by solving the two-dimensional reactive Navier–Stokes equations with an AMR to resolve important events in the flow. The computations show three stages of flame acceleration and how they are affected by the inclination of the array. The initial stage is the flame acceleration due to the increase of the flame surface area by the flow around the cylinders. The entrainment of the flame in the wake of the cylinders contributes to the fastest flame acceleration for the inclination angle 0° at which the row of cylinders is parallel to the flame propagation. As the inclination angle varies, formation of funnels of unburned material in the wake becomes dominant, and the flame does not accelerate as quickly. In the second stage, shock-flame interactions are responsible for the flame acceleration. The high reaction rate in the shock-compressed material decreases the flame surface area, while sustaining the energy-release rate. The final stage is a quasi-steady state of supersonic propagation. Detonation ignited by a local explosion propagates as a quasi-detonation. For 0° inclination, the local explosion does not lead to detonation, and the flame propagates in the choking regime.

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

SUMMARYThis paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.

Steady Combustion Waves Driven by a Recombination Reaction in a Gas Mixture

Steady combustion waves in one space dimension are studied by means of fluid-dynamic reactive Navier–Stokes equations derived from kinetic theory. By using the conservation laws, the problem is reduced to a three-dimensional dynamical system, with different solutions of deflagration or detonation type according to several various sub-regions in the parameter space. The delicate question of the flame eigenvalue is addressed, and a suitable numerical algorithm is used in order to catch these special solutions and to deal with the relevant unstable equilibrium states. Numerical examples of smooth weak deflagrations or detonations are shown.

Flame acceleration and transition to detonation in an array of square obstacles

We study flame acceleration and DDT in a two-dimensional staggered array of square obstacles by solving the compressible multidimensional reactive Navier–Stokes equations. The energy release rate for a stoichiometric H2-air mixture is modeled by a one-step Arrhenius kinetics. The space between obstacles is filled with a stoichiometric H2-air mixture at 1 atm and 298 K. Initially, the flow is at rest, and a flame is ignited at the center of the array. Computations show effects of the obstacles as a series of events leading to DDT. During the initial flame acceleration, the speed of the flame depends on the direction of flame propagation since some directions are more obstructed than others. This affects the macroscopic shape of the expanding burned region, which forms concave boundaries in more obstructed directions. As the flame accelerates, shocks form ahead of the flame, reflect from obstacles, and interact with the flame. There are more shock–flame interactions in more obstructed directions, and this leads to a greater flame acceleration and stronger leading shocks. When the shocks become strong enough, their collisions with obstacles ignite the gas mixture, and detonations form. The simulation shows four independent DDT events within a 90-degree sector, all in more obstructed directions. Resulting detonations spread in all directions. Some parts of detonation fronts are quenched by diffractions around obstacles, but they are reignited by collisions of decoupled shocks, or overtaken by other detonations. Thus detonations continue to spread and quickly burn all the material between the obstacles.

An adaptive high-order hybrid scheme for compressive, viscous flows with detailed chemistry

A hybrid weighted essentially non-oscillatory (WENO)/centered-difference numerical method, with low numerical dissipation, high-order shock-capturing, and structured adaptive mesh refinement (SAMR), has been developed for the direct numerical simulation of the multicomponent, compressible, reactive Navier–Stokes equations. The method enables accurate resolution of diffusive processes within reaction zones. The approach combines time-split reactive source terms with a high-order, shock-capturing scheme specifically designed for diffusive flows. A description of the order-optimized, symmetric, finite difference, flux-based, hybrid WENO/centered-difference scheme is given, along with its implementation in a high-order SAMR framework. The implementation of new techniques for discontinuity flagging, scheme-switching, and high-order prolongation and restriction is described. In particular, the refined methodology does not require upwinded WENO at grid refinement interfaces for stability, allowing high-order prolongation and thereby eliminating a significant source of numerical diffusion within the overall code performance. A series of one-and two-dimensional test problems is used to verify the implementation, specifically the high-order accuracy of the diffusion terms. One-dimensional benchmarks include a viscous shock wave and a laminar flame. In two-space dimensions, a Lamb–Oseen vortex and an unstable diffusive detonation are considered, for which quantitative convergence is demonstrated. Further, a two-dimensional high-resolution simulation of a reactive Mach reflection phenomenon with diffusive multi-species mixing is presented.

Hydrogen-oxygen flame acceleration and transition to detonation in channels with no-slip walls for a detailed chemical reaction model

The features of flame acceleration in channels with wall friction and the deflagration to detonation transition (DDT) are investigated theoretically and using high resolution numerical simulations of two-dimensional reactive Navier-Stokes equations, including the effects of viscosity, thermal conduction, molecular diffusion, and a detailed chemical reaction mechanism for hydrogen-oxygen gaseous mixture. It is shown that in a wide channel, from the beginning, the flame velocity increases exponentially for a short time and then flame acceleration decreases, ending up with the abrupt increase of the combustion wave velocity and the actual transition to detonation. In a thin channel with a width smaller than the critical value, the exponential increase of the flame velocity is not bounded and ends up with the transition to detonation. The transition to detonation occurs due to the pressure pulse, which is formed at the tip of the accelerating flame. The amplitude of the pressure pulse grows exponentially due to a positive feedback coupling between the pressure pulse and the heat released in the reaction. Finally, large amplitude pressure pulse steepens into a strong shock coupled with the reaction zone forming the overdriven detonation. The evolution from a temperature gradient to a detonation via the Zeldovich gradient mechanism and its applicability to the deflagration-to-detonation transition is investigated for combustible materials whose chemistry is governed by chain-branching kinetics. The results of the high resolution simulations are fully consistent with experimental observations of the flame acceleration and DDT.

Flame acceleration and DDT of hydrogen–oxygen gaseous mixtures in channels with no-slip walls

Hydrogen–oxygen flame acceleration and transition from deflagration to detonation (DDT) in channels with no-slip walls were studied theoretically and using high resolution simulations of 2D reactive Navier–Stokes equations, including the effects of viscosity, thermal conduction, molecular diffusion, real equation of state and a detailed chemical reaction mechanism. It is shown that in “wide” channels ( D > 1 mm) there are three distinctive stages of the combustion wave propagation: the initial short stage of exponential acceleration; the second stage of slower flame acceleration; the third stage of the actual transition to detonation. In a thin channel ( D < 1 mm) the flame exponential acceleration is not bounded till the transition to detonation. While velocity of the steady detonation waves formed in wider channels (10, 5, 3, 2 mm) is close to the Chapman–Jouguet velocity, the oscillating detonation waves with velocities slightly below the CJ velocity are formed in thinner channels ( D < 1.0 mm). We analyse applicability of the gradient mechanism of detonation ignition for a detailed chemical reaction model to be a mechanism of the deflagration-to-detonation transition. The results of high resolution simulations are fully consistent with experimental observations of flame acceleration and DDT in hydrogen–oxygen gaseous mixtures.

Detonation–turbulence interaction

A numerical study is performed to investigate the effect of preshock turbulence on the detonation wave properties. The analysis is based on the integration of the chemically reactive Navier–Stokes equations using a Runge–Kutta scheme and a fifth-order WENO spatial discretization. The results show a marked influence of preshock perturbations on the postshock statistics. The alteration to the limit cycle structure supported by unstable waves close to their critical points is highlighted. The effect of reactivity and fluid acceleration in the postshock region are examined by comparison with the non-reactive analog. The possibility that significant forcing can lead to hot spot formation is investigated by considering temperature probability distribution functions in the reaction zone. The separate effect of vortical and entropic fluctuations is considered.

Using staggered grids with characteristic boundary conditions when solving compressible reactive Navier–Stokes equations

AbstractA specific (hybrid) arrangement of variables is discussed to solve reactive compressible Navier–Stokes equations on staggered‐like grids with high‐order finite difference schemes. The objective is to improve the numerical flow solution at boundaries. Hybrid arrangement behaviour is compared with ‘pure’ colocated and staggered strategies. Classical Fourier analysis shows accuracy to be significantly improved in the hybrid case. One‐dimensional laminar flame test demonstrates increased robustness (in terms of mesh resolution), whereas computation of 1D exiting pressure wave propagation gives evidence that the method also improves accuracy in the prediction of non‐reflecting outflows, compared e.g. with the fully staggered scheme of (J. Comput. Phy. 2003). Multidimensional extension is illustrated through turbulent 2D planar and 3D expanding flames simulations. Copyright © 2011 John Wiley &amp; Sons, Ltd.

A Runge–Kutta discontinuous Galerkin approach to solve reactive flows: The hyperbolic operator

A Runge–Kutta discontinuous Galerkin method to solve the hyperbolic part of reactive Navier–Stokes equations written in conservation form is presented. Complex thermodynamics laws are taken into account. Particular care has been taken to solve the stiff gaseous interfaces correctly with no restrictive hypothesis. 1D and 2D test cases are presented.

Deflagration-to-detonation transition in highly reactive combustible mixtures

The paper presents experimental, theoretical, and numerical studies of deflagration-to-detonation transition (DDT) in highly reactive hydrogen–oxygen and ethylene–oxygen mixtures. Two-dimensional reactive Navier–Stokes equations for a hydrogen–oxygen gaseous mixture including the effects of viscosity, thermal conduction, molecular diffusion, and a detailed chemical reaction mechanism are solved numerically. It is found that mechanism of DDT is entirely determined by the features of the flame acceleration in tubes with no-slip walls. The experiments and computations show three distinct stages of the process: (1) the flame accelerates exponentially producing shock waves far ahead from the flame, (2) the flame acceleration decreases and shocks are formed directly on the flame surface, and (3) the final third stage of the actual transition to a detonation. During the second stage a compressed and heated pocket of unreacted gas adjacent ahead to the flame—the preheat zone is forming and the compressed unreacted mixture entering the flame produces large amplitude pressure pulse. The increase of pressure enhances reaction rate and due to a positive feedback between the pressure peak and the reaction the pressure peak grows exponentially, steepens into a strong shock that is coupled with the reaction zone forming the overdriven detonation wave. The proposed new physical mechanism of DDT highlights the features of flame acceleration in tubes with no-slip walls, which is the key factor of the DDT origin.

Heteroclinic orbits with fast transitions: a new construction of detonation profiles

We give a direct and elementary construction of strong detonation for the reactive Navier-Stokes equations (RNS), starting with an template given by a solution to the Zeldovitch-von Neumann-Doring (ZND) equations. Assuming that the viscosity, heat conductivity, and species diffusion coefficients in RNS are all proportional to e, we construct detonation ω e (x) that are exact solutions of RNS which converge in an appropriate sense to the given ZND profile as e → 0. The construction is explicit in the sense that it produces an arbitrarily high order expansion in powers of e for ω e , and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. Moreover, the leading term in the expansion is the original ZND profile, and the burned and unburned endstates of each RNS profile ω e coincide with those of the given ZND profile. The method used here is applicable to a variety of singular perturbation problems in which one seeks to construct smooth viscous profiles, involving both slow and fast transition regions, that converge to discontinuous inviscid profiles as a viscosity parameter e tends to 0. The method is applicable, for example, to second-order systems that cannot be written in conservative form, and can be used to construct solutions with fast transitions in situations, like two-point boundary problems, where no rest points or higher dimensional invariant manifolds are present in the reduced problem.

Singularly perturbed ODEs and profiles for stationary symmetric Euler and Navier-Stokes shocks

We construct stationary solutions to the non-barotropic, compressible Euler and Navier-Stokes equations in several space dimensions with spherical or cylindrical symmetry. The equation of state is assumed to satisfy standard monotonicity and convexity assumptions. For given Dirichlet data on a sphere or a cylinder we first construct smooth and radially symmetric solutions to the Euler equations in an exterior domain. On the other hand, stationary smooth solutions in an interior domain necessarily become sonic and cannot be continued beyond a critical inner radius. We then use these solutions to construct entropy-satisfying shocks for the Euler equations in the region between two concentric spheres (or cylinders). Next we construct smooth solutions w e to the Navier-Stokes system converging to the previously constructed Euler shocks in the small viscosity limit e → 0. The viscous solutions are obtained by a new technique for constructing solutions to a class of two-point boundary problems with a fast transition region. The construction is explicit in the sense that it produces high order expansions in powers of e for w e , and the coefficients in the expansion satisfy simple, explicit ODEs, which are linear except in the case of the leading term. The solutions to the Euler equations described above provide the slowly varying contribution to the leading term in the expansion. The approach developed here is applicable to a variety of singular perturbation problems, including the construction of heteroclinic orbits with fast transitions. For example, a variant of our method is used in [W] to give a new construction of detonation profiles for the reactive Navier-Stokes equations.

Flame structure from a kinetic model for chemical reactions

Steady one-dimensional flame structure is investigated in a binary mixture made up by two components of the same chemical speciesundergoing binary irreversible exothermic reactive encounters. A kinetic model at the Boltzmann level,accounting for chemical transitions as well as for mechanical collisions, is proposed and its main features are analyzed.In the case of slow chemical reactions and collision dominated regime, the model is the starting point for a consistent derivation,via suitable asymptotic expansion of Chapman-Enskog type, of reactive Navier-Stokes equations at the fluid-dynamic scale.The resulting set of ordinary differential equations is investigated in the frame of the qualitative theory of dynamical systems,and numerical results are presented and briefly commented on for illustrative purposes.