High resolution numerical simulation of the structure of 2-D gaseous detonations

Two dimensional numerical simulation of the structure of gaseous detonation is investigated by utilizing the single step Arrhenius kinetic reaction mechanism in both high and low activation energy mixtures, characterized by their irregular and regular detonation structure, respectively. All the computations are performed on a small Beowulf cluster with six nodes. The dependency of the structure on the grid resolution is performed and it is found that, resolution of more than 300 cells per hrl is required to demonstrate the role of hydrodynamic instabilities, (KH and RM instabilities) in detonation propagation in irregular structures, while due to the absence of fine-scale structures, resolution of 50 cells per hrl, gives the physical structure of detonation with regular structures. Results show that the transverse waves in irregular structure are significantly stronger than the transverse wave in regular structure detonation, which can enhance the burning rate of the unburned pockets behind the shock front. Results for resolution of 600 cells per hrl illustrate that, in addition to the primary mode, the interaction of large vortices with the shock front provides secondary modes in the structure which leads to the irregularity of the structure in high activation energy mixture. In contrast with the results obtained for regular structure, which no unburned gas pockets and vortices observed behind the front, the results for irregular structure reveal that most portions of the gases, escape from shock compression and create large unburned gas pockets behind the both weak section of the Mach stem and the incident wave, which will burn eventually by the turbulent mixing due to the vortices associated with hydrodynamic instabilities. Therefore, the ignition mechanism in irregular structure is due to the both shock compression and by turbulent mixing associated with hydrodynamic instabilities, while the shock compression yields the ignition mechanism in regular structure detonation.     

Computational chaos in nonlinear optics

A few models of nonlinear optical systems, known experimentally to possess both stable and unstable dynamical modes, are approximated by different dynamical models and integrated by different numerical methods. It is shown that the onset of instabilities and chaotic behavior in the same physical system may be dependent on the model used and on the numerical method applied. Finite order difference schemes should be applied with caution to infinite dimensional dynamical systems displaying irregular behavior.     

激波作用不同椭圆氦气柱过程中流动混合研究

在激波与气柱相互作用问题中,压力与密度间断不平行产生的斜压涡量会引起流动的不稳定性,从而促进物质间的混合.本文基于双通量模型,结合五阶加权基本无振荡(WENO)格式,求解多组分二维Navier-Stokes方程,分析激波作用面积相同结构不同的椭圆气柱所致的流动和混合.数值结果清晰地显示了激波诱导Richtmyer-Meshkov不稳定性引起的气柱界面变形和波系演化.同时定量地从界面运动、界面结构参数变化(长度和高度)、气柱体积压缩率、环量及混合率等角度分析激波诱导的流动混合机制,研究椭圆几何构型对氦气混合过程的影响.结果表明,界面及相关参数的演化与气柱初始形状密切相关.当激波沿椭圆长轴作用于气柱时,气柱前端出现空气射流结构,且射流不断增长并渗透到下游界面,致使气柱分离成两个独立涡团,离心率越大,射流发展越快;同时激波作用气柱后在界面处产生不规则反射现象.圆形气柱界面演化与这种作用情形类似.当激波沿椭圆短轴作用于气柱时,界面上游出现类平面结构,随后平面上下缘处产生涡旋,主导流动发展,激波在界面作用产生规则反射,离心率越大,这些现象越明显.界面高度、长度、体积压缩率也因此有所差异.对界面演化、环量和混合率的综合分析表明,激波沿长轴作用于气柱且离心率较大时,流动发展较快,不稳定性导致的流动越复杂,越有利于氦气与环境介质的混合.     

Numerical prediction of interfacial instabilities: Sharp interface method (SIM)

We introduce a sharp interface method (SIM) for the direct numerical simulation of unstable fluid–fluid interfaces. The method is based on the level set approach and the structured adaptive mesh refinement technology, endowed with a corridor of irregular, cut-cell grids that resolve the interfacial region to third-order spatial accuracy. Key in that regard are avoidance of numerical mixing, and a least-squares interpolation method that is supported by irregular datasets distinctly on each side of the interface. Results on test problems show our method to be free of the spurious current problem of the continuous surface force method and to converge, on grid refinement, at near-theoretical rates. Simulations of unstable Rayleigh–Taylor and viscous Kelvin–Helmholtz flows are found to converge at near-theoretical rates to the exact results over a wide range of conditions. Further, we show predictions of neutral-stability maps of the viscous Kelvin–Helmholtz flows (Yih instability), as well as self-selection of the most unstable wave-number in multimode simulations of Rayleigh–Taylor instability. All these results were obtained with a simple seeding of random infinitesimal disturbances of interface-shape, as opposed to seeding by a complete eigenmode. For other than elementary flows the latter would normally not be available, and extremely difficult to obtain if at all. Sample comparisons with our code adapted to mimic typical diffuse interface treatments were not satisfactory for shear-dominated flows. On the other hand the sharp dynamics of our method would appear to be compatible and possibly advantageous to any interfacial flow algorithm in which the interface is represented as a discrete Heaviside function.     

The evolution model of the rayleigh-taylor instability development

The turbulent mixing (TM) of different DT-fuel areas (cold with hot) and of DT-fuel with ablator is one of the deciding factors, determining the neutron yield from compressing laser fusion, and, more generally, inertial confinement fusion (ICF) targets. A lower than predicted neutron yield gained in experiments is not studied comprehensively, but can be caused by the mixing processes. A study of mixing in the ICF problem is complicated by density gradients, spherical convergence, compression, etc., so we suppose the fundamental understanding of the mixing processes must be acquired first in the problem of classical Rayleigh-Taylor (RT) instability in plain geometry. In our work we present results, obtained by the supercomputer numerical modeling of RT-induced TM processes with different initial conditions in 2D and 3D geometries conditions on a high-resolution meshes. For analysis of the modeling results we use an evolutionary model of singular perturbation development, including linear and non-linear stages. This theoretical approach allows us to obtain highly detailed view of the mixing zone evolution along with possibility to trace the initial conditions influence on the mixing late stages.     

Mixing time analysis of a sonochemical reactor

Mixing time measurements have been carried out in a cylindrical reactor irradiated with an ultrasonic horn fitted with different size tips. Liquid phase bulk velocities induced by the vibrating horn surface have been estimated from the mixing time measurements. A relationship has been established between the mean horn surface velocities (frequency x amplitude) and the mean velocities estimated from the mixing time measurements. A correlation has been developed for the prediction of the mixing time using a method similar to that used for liquid jet mixing. This could be the first step in defining the overall flow field, the information about which can then be used to get realistic numerical solutions of the Rayleigh-Plesset equation for a travelling cavity to understand the cavity dynamics in the various parts of the ultrasonic horn reactor.     

Coupling of mixing models with manifold based simplified chemistry in PDF modeling of turbulent reacting flows

For general reacting flows the numerical simulation faces two main challenges. One is the high dimensionality and stiffness of the governing conservation equations due to detailed chemistry, which can be solved by using simplified chemical kinetics. The other one is the difficulty of modeling the coupling of turbulence with thermo-chemical source term. The probability density function (PDF) method allows to calculate turbulent reacting flows by solving the thermal-chemical source term in closed form. Usually, the PDF method for turbulent processes such as mixing processes and the reduction method for chemical kinetics are developed separately. However, coupling of both processes plays an important role for the numerical accuracy. To investigate the importance of coupling between turbulence and simplified chemistry, two different coupling strategies for mixing and reduced chemistry are discussed and tested for the well-known Sandia Flames E and F, in which there is a strong interaction between turbulence and chemical kinetics. The EMST mixing model is chosen for turbulent mixing, while the Reaction-Diffusion Manifolds (REDIMs) is used as simplified chemistry. However, the proposed strategies are also valid for other mixing models and manifold based simplified chemistry.     

Pulse propagation in four-wave mixing process with group index difference of signal and idler waves

Four-wave mixing process with a large group index difference of the signal and idler pulses in a nonlinear optical fiber is theoretically investigated when the pump is continuous light. We prove that in the four-wave mixing process, the signal and idler waves would finally propagate in a common group velocity in spite of their different group indices. When the effective phase mismatch in four-wave mixing is not zero, their carrier frequencies shift in different directions. The asymptotic result of the signal and idler shape and carrier frequency shift are obtained. The theoretical prediction is validated by the numerical simulation.     

Rayleigh-Taylor turbulence is nothing like Kolmogorov turbulence in the self-similar regime

An increasing number of numerical simulations and experiments describing the turbulent spectrum of Rayleigh-Taylor (RT) mixing layers came to light over the past few years. Results reported in recent studies allow to rule out a Kolmogorov-like turbulence as a mechanism acting on a self-similar RT turbulent mixing layer. A different mechanism is presented, which complies with both numerical and experimental results and relates RT flow to other buoyant flows.     

A low numerical dissipation immersed interface method for the compressible Navier–Stokes equations

A numerical method to solve the compressible Navier–Stokes equations around objects of arbitrary shape using Cartesian grids is described. The approach considered here uses an embedded geometry representation of the objects and approximate the governing equations with a low numerical dissipation centered finite-difference discretization. The method is suitable for compressible flows without shocks and can be classified as an immersed interface method. The objects are sharply captured by the Cartesian mesh by appropriately adapting the discretization stencils around the irregular grid nodes, located around the boundary. In contrast with available methods, no jump conditions are used or explicitly derived from the boundary conditions, although a number of elements are adopted from previous immersed interface approaches. A new element in the present approach is the use of the summation-by-parts formalism to develop stable non-stiff first-order derivative approximations at the irregular grid points. Second-order derivative approximations, as those appearing in the transport terms, can be stiff when irregular grid points are located too close to the boundary. This is addressed using a semi-implicit time integration method. Moreover, it is shown that the resulting implicit equations can be solved explicitly in the case of constant transport properties. Convergence studies are performed for a rotating cylinder and vortex shedding behind objects of varying shapes at different Mach and Reynolds numbers.     

Using the shooting method to solve boundary-value problems involving nonlinear coupled-wave equations

Using the shooting method, one of the numerical methods to solve two-point boundary-value problems, numerical solutions of the nonlinear coupled-wave equations in degenerate two-wave and four-wave mixing can be obtained. In this first part of the paper the general shooting method is described, and then applied to two-wave mixing in a reflection geometry. Computed results are presented in graphical form. Comparison between the shooting method and the direct numerical method [1] is made also. In the second part of the paper, numerical solutions for four-wave mixing in a reflection geometry will be given.     

On the dynamics of parabolic equations describing diffusion, convection and a chemical reaction

A tubular nonisothermal-nonadiabatic chemical reactor with a consecutive reactions described by a set of three nonlinear parabolic equations, shows a sequence of period-doubling bifurcations of a “limit cycle”. A numerical examination of the model reveals that complex periodic and irregular oscillations are possible. With increasing value of the Peclet number regular and irregular oscillations are suppressed and disappear. From the computed results may be informed that for reacting systems nonlinear parabolic equations may feature similar qualitative properties as the ordinary differential equations. The results of this numerical study may be used to explain turbulization of laminar flames.     

随机分布黑碳-硅酸盐混合凝聚粒子的消光特性研究

基于分形理论,采用蒙特卡罗方法对随机分布的混合凝聚粒子的空间结构进行了仿真模拟。利用Bruggeman有效介质理论得到了占有不同体积份额黑碳的内混合凝聚粒子的等效复折射率。采用离散偶极子近似方法对随机分布混合凝聚粒子在内外混合状态下的吸收、散射和消光效率因子等消光特性参量进行了数值计算,深入探讨了混合方式、容积含量、入射波长以及基本粒子粒径和数量对混合凝聚粒子消光特性的影响规律。通过将所得数值结果与T矩阵方法的数值结果进行比较发现,两种数值方法计算的结果非常相近。结果表明,随机分布混合凝聚粒子的散射效率因子对混合方式非常敏感,消光效率因子对混合方式较敏感,而吸收效率因子对混合方式不敏感。随着凝聚粒子尺度参数的增大,混合方式对散射和消光效率因子的影响逐渐显著。内外混合方式下,随着黑碳体积比的增大随机分布混合凝聚粒子的吸收、散射和消光效率因子均近似线性增大,并且增大的幅度随着粒子尺度参数的增大而增大。     

Numerical simulation of the hydrodynamic instability experiments and flow mixing

Based on the numerical methods of volume of fluid (VOF) and piecewise parabolic method (PPM) and parallel circumstance of Message Passing Interface (MPI), a parallel multi-viscosity-fluid hydrodynamic code MVPPM (Multi-Viscosity-Fluid Piecewise Parabolic Method) is developed and performed to study the hydrodynamic instability and flow mixing. Firstly, the MVPPM code is verified and validated by simulating three instability cases: The first one is a Riemann problem of viscous flow on the shock tube; the second one is the hydrodynamic instability and mixing of gaseous flows under re-shocks; the third one is a half height experiment of interfacial instability, which is conducted on the AWE’s shock tube. By comparing the numerical results with experimental data, good agreement is achieved. Then the MVPPM code is applied to simulate the two cases of the interfacial instabilities of jelly models accelerated by explosion products of a gaseous explosive mixture (GEM), which are adopted in our experiments. The first is implosive dynamic interfacial instability of cylindrical symmetry and mixing. The evolving process of inner and outer interfaces, and the late distribution of mixing mass caused by Rayleigh-Taylor (RT) instability in the center of different radius are given. The second is jelly layer experiment which is initialized with one periodic perturbation with different amplitude and wave length. It reveals the complex processes of evolution of interface, and presents the displacement of front face of jelly layer, bubble head and top of spike relative to initial equilibrium position vs. time. The numerical results are in excellent agreement with that experimental images, and show that the amplitude of initial perturbations affects the evolvement of fluid mixing zone (FMZ) growth rate extremely, especially at late times.     

Small-scale anisotropy in turbulent shearless mixing

The generation of small-scale anisotropy in turbulent shearless mixing is numerically investigated. Data from direct numerical simulations at Taylor Reynolds' numbers between 45 and 150 show not only that there is a significant departure of the longitudinal velocity derivative moments from the values found in homogeneous and isotropic turbulence but that the variation of skewness has an opposite sign for the components across the mixing layer and parallel to it. The anisotropy induced by the presence of a kinetic energy gradient has a very different pattern from the one generated by an homogeneous shear. The transversal derivative moments in the mixing are in fact found to be very small, which highlights that smallness of the transversal moments is not a sufficient condition for isotropy.     

Numerical models for the study of the nonlinear frequency mixing in two and three-dimensional resonant cavities filled with a bubbly liquid

The objective of this work is to develop versatile numerical models to study the nonlinear distortion of ultrasounds and the generation of low-ultrasonic frequency signals by nonlinear frequency mixing in two and three-dimensional resonators filled with bubbly liquids. The interaction of the acoustic field and the bubble vibrations is modeled through a coupled differential system formed by the multi-dimensional wave equation and a Rayleigh-Plesset equation. The numerical models we develop are based on multi-dimensional finite-volume techniques and a time discretization carried out by finite differences. Numerical experiments are performed for complex modes in many different cavities considering different kinds of boundary conditions and taking advantage of the dispersive character of the bubbly fluid to match specific resonances of the cavities. Results show the distribution of fundamental and harmonics for single frequency excitation and difference-frequency component for two-frequency excitation that are promoted by the strong nonlinearity of the bubbly medium. The numerous simulations analyzed suggest that the new numerical models developed and proposed in this paper are useful to understand the behavior of ultrasounds in bubbly liquids for sonochemical processes and applications of nonlinear frequency mixing.     

Oscillations vs. chaotic waves: Attractor selection in bistable stochastic reaction–diffusion systems

In bistable systems, the long-term behavior of solutions depends on the location of the initial conditions. In a deterministic setting, where the initial condition is kept fixed in one particular basin of attraction, repeated numerical simulations will always lead to the same long-term behavior. The other possible asymptotic solution type will never be observed. This clear distinction does not hold anymore if the system is forced by random fluctuations. In this case, both asymptotic solutions can be attained, and the relative frequency of different long-term behaviors observed in many repeated simulation runs will follow a certain probability distribution. We present a simple reaction–diffusion model of spatial predator–prey interaction, where depending on the initial spatial distribution of the two populations either spatially homogeneous or spatiotemporal irregular oscillations may be observed. We show by repeated stochastic simulations that, when starting in the basin of attraction of the spatiotemporal irregular solution, in the randomly forced system the probability to observe spatially homogeneous oscillations instead of spatiotemporally irregular oscillations follows a non-trivial bimodal distribution.     

二维拉格朗日坐标系下气粒混合双向耦合对激波流场影响的计算

喷射颗粒与气体混合是内爆压缩领域的热点和难点. 针对喷射混合中的气粒双向耦合问题, 开展了理论建模、离散算法以及颗粒反馈对激波流场的影响研究. 建立了拉格朗日计算框架下的数学模型; 给出了耦合源项的离散算法; 开展了平面及汇聚构型条件下, 气粒双向耦合的数值模拟研究; 发现了颗粒反馈导致气体激波提速现象以及气区流场物理量分布形态的改变, 初步获得了量化分析结果. 本文建立的数学模型、计算方法和获得的新的物理认识, 为深入理解喷射混合现象、解决相关工程应用问题提供了重要理论支撑.     

Efficiency of pseudo-spectral algorithms with Anderson mixing for the SCFT of periodic block-copolymer phases

This study examines the numerical accuracy, computational cost, and memory requirements of self-consistent field theory (SCFT) calculations when the diffusion equations are solved with various pseudo-spectral methods and the mean-field equations are iterated with Anderson mixing. The different methods are tested on the triply periodic gyroid and spherical phases of a diblock-copolymer melt over a range of intermediate segregations. Anderson mixing is found to be somewhat less effective than when combined with the full-spectral method, but it nevertheless functions admirably well provided that a large number of histories is used. Of the different pseudo-spectral algorithms, the 4th-order one of Ranjan, Qin and Morse performs best, although not quite as efficiently as the full-spectral method.