Some mathematical aspects of flame chaos and flame multiplicity

In these review lectures, we survey recent mathematical results on the low, finite-dimensional behavior of the Kuramoto-Sivashinsky equations; the latter model thermo-diffusive flame front instabilities. We outline how these PDE's are rigorously equivalent to a finite dynamical system (ongoing joint work with C. Foias, G. Sell, R. Temam). We also present an introduction to multiplicity phenomena in flames caused by chemical kinetics (the latter is an ongoing joint work with P. Clavin and P.C. Fife).     

Nonlinear simulations of combustion instabilities with a quasi-1D Navier-Stokes code

As lean premixed combustion systems are more susceptible to combustion instabilities than non-premixed systems, there is an increasing demand for improved numerical design tools that can predict the occurrence of combustion instabilities with high accuracy. The inherent nonlinearities in combustion instabilities can be of crucial importance, and we here propose an approach in which the one-dimensional (1D) Navier-Stokes and scalar transport equations are solved for geometries of variable cross-section. The focus is on attached flames, and for this purpose a new phenomenological model for the unsteady heat release from a flame front is introduced. In the attached flame method (AFM) the heat release occurs over the full length of the flame. The nonlinear code with the use of the AFM approach is validated against analytical results and against an experimental study of thermoacoustic instabilities in oxy-fuel flames by Ditaranto and Hals [Combustion and Flame 146 (2006) 493-512]. The numerical simulations are in accordance with the experimental measurements and the analytical results and both the frequencies and the amplitudes of the resonant acoustic pressure modes are reproduced with good accuracy.     

An ignition-temperature model with two free interfaces in premixed flames†

In this paper we consider an ignition-temperature zero-order reaction model of thermo-diffusive combustion. This model describes the dynamics of thick flames, which have recently received considerable attention in the physical and engineering literature. The model admits a unique (up to translations) planar travelling wave solution. This travelling wave solution is quite different from those usually studied in combustion theory. The main qualitative feature of this travelling wave is that it has two interfaces: the ignition interface where the ignition temperature is attained and the trailing interface where the concentration of deficient reactants reaches zero. We give a new mathematical framework for studying the cellular instability of such travelling front solutions. Our approach allows the analysis of a free boundary problem to be converted into the analysis of a boundary value problem having a fully nonlinear system of parabolic equations. The latter is very suitable for both mathematical and numerical analysis. We prove the existence of a critical Lewis number such that the travelling wave solution is stable for values of Lewis number below the critical one and is unstable for Lewis numbers that exceed this critical value. Finally, we discuss the results of numerical simulations of a fully nonlinear system that describes the perturbation dynamics of planar fronts. These simulations reveal, in particular, some very interesting ‘two-cell’ steady patterns of curved combustion fronts.     

基于Spalart-Allmaras-γ-\begin{document} ${\overline {Re} _{\theta {\rm{t}}}}$ \end{document}转捩模型的横流不稳定性转捩预测方法

传统的应用稳定性理论对横流不稳定性转捩现象的预测很难与现代CFD并行化计算结合, 为了解决这个问题, 文章基于SA-γ-${\overline {Re} _{\theta {\rm{t}}}}$转捩模型, 使用不可压三维边界层相似性解实现横流位移损失厚度Reynolds数在流场中的当地化求解, 结合亚音速试验数据-C1准则构建横流不稳定性转捩判据, 从而实现了横流不稳定性转捩预测方法的当地化并行求解.首先采用SA-γ-${\overline {Re} _{\theta {\rm{t}}}}$转捩模型对NLF(1)-0416翼型进行了流向转捩预测, 证实了该模型的正确性.然后应用所建立的横流转捩模型对45°前缘后掠角的NLF(2)-0415无限展长机翼和DLR-F5机翼, 以及标准6:1椭球标模进行了横流不稳定转捩数值模拟, 计算结果显示转捩位置均与试验数据吻合较好, 证明了文章所建立的方法在不可压边界层转捩预测具有较高的预测精度.      

Cellular and sporadic flame regimes of low-Lewis-number stretched premixed flames

3D structure and dynamical behavior of low-Lewis-number stretched premixed flames are numerically simulated within the framework of a thermo-diffusive model with one-step chemical reaction. The results are compared with microgravity experiments at qualitative level. The influence of Lewis number, equivalence ratio, and heat loss intensity on flame structure is investigated. It is experimentally and numerically found that lean counterflow flames can appear as a set of separate ball-like flames in a state of chaotic motion. It is shown that the time averaged flame balls coordinate may be considered as important characteristic similar to coordinate of continuous flame front. Numerical simulations reveal essential incompleteness of combustion at high level of heat losses. This incompleteness occurs in the process of lean mixtures combustion and is caused by fuel leakage through the gaps among ball-like flames.     

The effect of hydrodynamical instabilities and of convection in the conversion of an hadronic star to a more compact object

We study the hydrodynamical transition from an hadronic star into a quark or a hybrid star. We discuss the possible mode of burning, using a fully relativistic formalism and realistic Equations of State in which hyperons can be present. We take into account the possibility that quarks form a diquark condensate. We find that the conversion process always corresponds to a deflagration and never to a detonation. Hydrodynamical instabilities can develop on the front but the increase in the conversion’s velocity is not sufficient to transform the deflagration into a detonation. Concerning convection, it does not always develop. Instead, the process of conversion from ungapped quark matter to gapped quarks always allows the formation of a convective layer. The text was submitted by the authors in English.     

On polyhedral structures of lean methane/hydrogen Bunsen flames: Combined experimental and numerical analysis

In premixed flame propagation of lean hydrogen or hydrogen-enriched blends, both hydrodynamic and thermo-diffusive instabilities are governing the flame front shape and affect its propagation velocity. As a result, different types of cellular patterns can occur along the flame front in a laminar scenario. In this context, an interesting phenomenon is the formation of polyhedral flames which can be observed in a Bunsen burner. It is the objective of this work to systematically characterize the polyhedral structures of premixed methane/hydrogen Bunsen flames in a combined experimental and numerical study. A series of lean flames with hydrogen content varying between 20 and 85% at two equivalence ratios is investigated. The experiments encompass chemiluminescence imaging together with Planar Laser-induced Fluorescence (PLIF) measurements of the OH radical. Characteristic cell sizes are quantified from the experiments and related to the characteristic length scales obtained from a linear stability analysis. In the experiments, it is observed that the cell sizes at the base of the polyhedral Bunsen flames decrease almost linearly with hydrogen addition and only a weak dependence on the equivalence ratio is noted. These trends are well reflected in the numerical results and the length scale comparison further shows that the wavelength with the maximum growth rate predicted by the linear stability analysis is comparable to the cell size obtained from the experiment. The correlation between the experimental findings and the linear stability analysis is discussed from multiple perspectives considering the governing time and length scales, furthermore drawing relations to previous studies on cellular flames.     

Stability of fronts separating domains with different symmetries in hydrodynamical instabilities

A generalization of the Swift-Hohenberg (SH) equation is used to study several stationary patterns that appear in hydrodynamical instabilities. The corresponding amplitude equations allow one to find the stability of planforms with different symmetries. These results are compared with numerical simulations of a generalized SH equation (GSHE). The transition between different symmetries, the hysteretic effects, and the characteristics of the defects observed in experiments are well reproduced in these simulations. The existence of steady fronts between domains with different symmetries is also analyzed. Steady domain boundaries between hexagons and rolls, and between hexagons and squares are possible solutions in the amplitude equation framework and are obtained in numerical simulations for a full range of coefficients in the GSHE.     

Instabilities of current distribution in smeared transition zones between two electrical conducting phases

In the transition zone between phases of different conduction types, the electrical current is not only unevenly distributed, but also, in many cases, local or temporal instabilities occur. Most of these effects have already been observed, but until now, no theoretical treatment has been published. As a straightforward extension of the linear mathematical treatment of the current distribution in porous electrodes¹, an explanation of two classes of purely electrical instabilities can be given. Beyond these purely electrical instabilities, a number of more complicated cases has to be considered. These are local instabilities caused by overlapping singularities, e.g. spots of minor density or conductivity, local and temporal instabilities caused by thermal breakthroughs and instabilities caused by hydrodynamical eddies. Moreover the current distribution shows local and temporal fluctuations as the result of the granular structure of the transition zone.     

使用槽道平均模型的多叶片排N-S方程并行计算

1前言多叶片排三元粘性计算，正在成为压气机、涡轮设计的重要工具。与混合平面模型山相比，槽道平均模型p］能够更加精确地模拟动静叶片间相互作用。它减少了内边界信息传递积累误差，增强了计算程序的鲁棒性。国内很多单位开展了有关的研究工作［‘－’］。本文发展了使用槽道平均模型多叶片N－S方程并行计算程序，并对一多级压气机设计中间方案的前3级进行了试算。2关于槽道周向平均模型这一模型是由Ad。mCZyk［’l提出，它不但被应用于三维直接计算数值模拟，其中的一些基本概念还曾被应用于两类流面迭代平均SZ流面计算。其主要思路…     

Subgrid modeling of intrinsic instabilities in premixed flame propagation

This work is devoted to the investigation and subgrid-scale modeling of intrinsic flame instabilities occurring in the propagation of a deflagration wave. Such instabilities, of hydrodynamic and thermodiffusive origin, are expected to be of particular relevance in recent technological trends such as in the use of hydrogen as a clean energy carrier or as a secondary fuel in hydrogen enriched combustion. A dedicated set of direct numerical simulations is presented and used, in conjunction with coherent literature results, in order to develop scaling arguments for the propagation speed of self-wrinkled flames which are also supported by the outcomes of a weakly non-linear model, namely the Sivashinsky equation. The observed scaling is based on the definition of the number of unstable wavelengths in a reference hydrodynamic lengthscale, in other words the ratio between the neutral or cutoff lengthscale of intrinsic instabilities and the lateral domain of a planar flame. The scalings are then employed to develop an algebraic model for the wrinkling factor in the context of a flame surface density closure approach. An a-priori analysis shows that the model correctly captures the flame wrinkling caused by intrinsic instability at sub grid level. A strategy to include the developed self-wrinkling model in the context of a turbulent combustion model is finally discussed on the basis of the turbulence induced cut-off concept.     

Electrostatic Instabilities at High Frequency in a Plasma Shock Front

New electrostatic instabilities in the plasma shock front are reported. These instabilities are driven by the electro- static field which is caused by charge separation and the parameter gradients in a plasma shock front. The linear analysis to the high frequency branch of electrostatic instabilities has been carried out and the dispersion relations are obtained numerically. There are unstable disturbing waves in both the parallel and perpendicular directions of shock propagation. The real frequencies of both unstable waves are similar to the electron electrostatic wave, and the unstable growth rate in the parallel direction is much greater than the one in the perpendicular direction. The dependence of growth rates on the electric field and parameter gradients is also presented.     

High-order algorithms for compressible reacting flow with complex chemistry

In this paper we describe a numerical algorithm for integrating the multicomponent, reacting, compressible Navier–Stokes equations, targeted for direct numerical simulation of combustion phenomena. The algorithm addresses two shortcomings of previous methods. First, it incorporates an eighth-order narrow stencil approximation of diffusive terms that reduces the communication compared to existing methods and removes the need to use a filtering algorithm to remove Nyquist frequency oscillations that are not damped with traditional approaches. The methodology also incorporates a multirate temporal integration strategy that provides an efficient mechanism for treating chemical mechanisms that are stiff relative to fluid dynamical time-scales. The overall methodology is eighth order in space with options for fourth order to eighth order in time. The implementation uses a hybrid programming model designed for effective utilisation of many-core architectures. We present numerical results demonstrating the convergence properties of the algorithm with realistic chemical kinetics and illustrating its performance characteristics. We also present a validation example showing that the algorithm matches detailed results obtained with an established low Mach number solver.     

A numerical modeling of the acceleration of a flame by an additional energy input ahead of its front

The acceleration of a flame after an additional energy input ahead of its front was simulated using numerical methods. The combustion of a hydrogen-air mixture in a semiopen channel was considered. The calculations were performed within the framework of a two-dimensional hydrodynamic model of premixed flames, with consideration given to heat transfer, multicomponent diffusion, and chemical kinetics. It was demonstrated that, when the interaction of the flame front with the near-wall boundary layer is taken into account, even a moderate energy input could substantially promote the development of the Landau-Darrieus instability and, possibly, deflagration to detonation transition.     

Impact of the gravity field on stability of premixed flames propagating between two closely spaced parallel plates

The stability of a planar flame front propagating between two parallel adiabatic plates inclined at an arbitrary angle is investigated in the frame of narrow-channel approximation. It is demonstrated that buoyancy forces can suppress the hydrodynamic (Darrieus–Landau) and cellular (diffusive-thermal) instabilities for sufficiently large value of the gravity parameter for the case of downward-propagating flames. The stability analysis reveals that in the case of oscillatory diffusive-thermal instability, the flame front cannot be stabilized in the similar way. Finally, the stability results are compared satisfactorily with unsteady numerical simulations.     

二维球坐标系中子输运方程的一种并行SN算法

针对二维球坐标系下中子输运方程的SN算法, 提出基于(单元, 方向)二元组的有向图模型, 在已有的基于有向图的并行流水线算法基础上, 设计粒度可控多级并行SN算法。其中, 采用区域分解和并行流水线相结合的方式挖掘空间-角度方向的并行度, 提出能群流水并行方法, 并通过设置合适的流水线粒度来平衡有向图调度、通信和空闲等待开销。实验结果表明: 该算法可以有效地求解二维球坐标系下的中子输运方程。在某国产并行机1920核上, 对于96万网格、60个方向、24能群、数十亿自由度的典型中子输运问题, 获得了71%的并行效率。     

基于GPU集群的Level Set并行高精度演化

设计实现基于张量积B样条的并行Level Set演化算法,张量积B样条提高了演化精度和并行度;每步演化都需要反算B样条系数.针对对角占优三对角方程组,设计实现基于精确LU分解的高精度并行追赶法,并用它反算B样条系数;采用两步通信方法,消除通信的依赖关系,实现有效的并行通信.实验表明,本文的并行算法可以有效加速演化过程.     

Pulsed modes of the formation of multilayer char structures on the surface of fire-retardant intumescent paints

A mathematical model of the propagation of the front of decomposition of active fire-retardant intumescent paints in pulsed mode is developed. A hypothesis explaining the mechanism of the experimen-tally observed pulsed modes of decomposition front propagation by the existence of an exothermic step in the decomposition reaction, leading to the self-acceleration of the reaction and rapid burnout of the reacting substance layer, is suggested. A theory of the propagation of the decomposition front in the pulsed mode based on a minimum number of empirical parameters obtainable from experiment is developed. Based on numerical simulation results, formulas are derived for predicting the time history of the thickness of the char structure and the time during which the fire-retardant composition can protect the object from fire. These formulas can be used to calculate the desired thickness of the fire-retardant coating that would withstand a fire or a thermal agent of given intensity for a desired time.