Simulation of shock-induced instability using an essentially conservative adaptive CE/SE method

An essentially conservative adaptive space time conservation element and solution element(CE/SE)method is proposed for the effective simulation of shock-induced instability with low computational cost.Its implementation is based on redefined conservation elements(CEs)and solution elements(SEs),optimized interpolations and a Courant number insensitive CE/SE scheme.This approach is used in two applications,the Woodward double Mach reflection and a twocomponent Richtmyer–Meshkov instability experiment.This scheme reveals the essential features of the investigated cases,captures small unstable structures,and yields a solution that is consistent with the results from experiments or other high order methods.     

高阶精度CE/SE算法及其应用

对时-空守恒元解元算法(CE/SE)的网格设置做较大改进,提出一种新的六面体解元和元定义;同时在解元中对物理量进行高阶Taylor展开,给出一种在时间和空间上均具有高阶精度CE/SE算法.在此基础上,把新型的高阶精度CE/SE算法推广应用于高速流动捕捉激波间断、气相化学反应流动、计及固体动态效应的流体-弹塑性流动和非稳态多相不可压缩粘性流动中.数值实践表明,提出的新型网格结构上的高阶精度CE/SE算法具有算法简单、计算精度高、计算效率和计算效果好的优点,并大大改进和拓展了CE/SE算法的应用范围.     

一种严格保证时-空守恒律的数值方法

对文[1]中的时-空守恒元和解元方法(CE/SE)进行了改进和发展。改变了原格式的构造方法,由此得到的新格式不仅保留了原方法的优点,而且格式更为简单,通用性更强,也更容易推广到高阶和多维情形。     

An Improved CE/SE Scheme and Its Application to Detonation Propagation

An improved two-dimensional space-time conservation element and solution element （CE/SE） method with second-order accuracy is proposed, examined and extended to simulate the detonation propagations using detailed chemical reaction models. The numerical results of planar and cellular detonation are compared with corresponding results by the Chapman-Jouguet theory and experiments, and prove that the method is a new reliable way for numerical simulations of detonation propagation.     

Acoustic characteristics of pulse detonation engine with ellipsoidal reflector

Acoustic characteristics of a pulse detonation engine(PDE) with and without an ellipsoidal reflector are numerically and experimentally investigated. A two-dimensional(2 D) non-splitting unstructured triangular mesh Euler solver based on the space-time conservation element and solution element(CE/SE) method is employed to simulate the flow field of a PDE.The numerical results clearly demonstrate the external flow field of the PDE. The effect of an ellipsoidal reflector on the flow field characteristic near the PDE exit is investigated. The formation process of reflected shock wave and reflected jet shock are reported in detail. An acoustic measurement system is established for the PDE acoustic testing. The experimental results show that the ellipsoidal reflector changes the sound waveform and directivity of PDE sound. The reflected shock wave and reflected jet shock result in two more positive pressure peaks in the sound waveform. The ellipsoidal reflector changes the directivity of PDE sound from 20 to 0. It is found that the peak sound pressure level(PSPL) and overall sound pressure level(OASPL) each obtain an increment when the PDE is installed with a reflector. The maximum relative increase ratio of PSPL and OASPL are obtained at the focus point F2, whose values are 6.1% and 6.84% respectively. The results of the duration of the PDE sound indicate that the reflecting and focusing wave generated by the reflector result in the increment of A duration and B duration before and near focus point F2. Results show that the ellipsoidal reflector has a great influence on the acoustic characteristic of PDE sound. The research is helpful for understanding the influence of an ellipsoidal reflector on the formation and propagation process of PDE sound.     

Three-dimensional detonation cellular structures in rectangular ducts using an improved CESE scheme

The three-dimensional premixed H_2-O_2 detonation propagation in rectangular ducts is simulated using an in-house parallel detonation code based on the second-order space-time conservation element and solution element(CE/SE) scheme.The simulation reproduces three typical cellular structures by setting appropriate cross-sectional size and initial perturbation in square tubes.As the cross-sectional size decreases,critical cellular structures transforming the rectangular or diagonal mode into the spinning mode are obtained and discussed in the perspective of phase variation as well as decreasing of triple point lines.Furthermore,multiple cellular structures are observed through examples with typical aspect ratios.Utilizing the visualization of detailed three-dimensional structures,their formation mechanism is further analyzed.     

A class of hybrid DG/FV methods for conservation laws I: Basic formulation and one-dimensional systems

By comparing the discontinuous Galerkin (DG) and the finite volume (FV) methods, a concept of ‘static reconstruction’ and ‘dynamic reconstruction’ is introduced for high-order numerical methods. Based on the new concept, a class of hybrid DG/FV schemes is presented for one-dimensional conservation law using a ‘hybrid reconstruction’ approach. In the hybrid DG/FV schemes, the lower-order derivatives of a piecewise polynomial solution are computed locally in a cell by the DG method based on Taylor basis functions (called as ‘dynamic reconstruction’), while the higher-order derivatives are re-constructed by the ‘static reconstruction’ of the FV method, using the known lower-order derivatives in the cell itself and its adjacent neighboring cells. The hybrid DG/FV methods can greatly reduce CPU time and memory required by the traditional DG methods with the same order of accuracy on the same mesh, and they can be extended directly to unstructured and hybrid grids in two and three dimensions similar to the DG and/or FV methods. The hybrid DG/FV methods are applied to one-dimensional conservation law, including linear and non-linear scalar equation and Euler equations. In order to capture the strong shock waves without spurious oscillations, a simple shock detection approach is developed to mark ‘trouble cells’, and a moment limiter is adopted for higher-order schemes. The numerical results demonstrate the accuracy, and the super-convergence property is shown for the third-order hybrid DG/FV schemes. In addition, by analyzing the eigenvalues of the semi-discretized system in one dimension, we discuss the spectral properties of the hybrid DG/FV schemes to explain the super-convergence phenomenon.     

拐角空间中悬浮铝粉尘爆轰后效的三维数值模拟

建立三维的铝粉-空气两相爆轰计算模型,采用时-空守恒元解元（CE/SE）方法求解,并开发了悬浮铝粉尘爆轰的三维数值模拟程序.基于消息传递接口（MPI）技术实现了程序的并行化设计.通过对激波管问题以及爆轰管中铝粉-空气两相爆轰实验的模拟验证程序的可靠性.对拐角空间中左侧浓度为368 g·m^-3的铝粉-空气混合物两相爆轰及其在拐角空间右侧和下方空气域内形成的冲击波和温压效应开展数值模拟,获得复杂空间内爆轰波或冲击波的传播、反射以及绕射过程.结果表明：两相爆轰在离铝粉尘区域2 m远的空气域内产生的后效冲击波能达到2.66 MPa的固壁反射压力,火球燃烧范围会超出初始铝粉尘区域约0.8 m,并且造成初始铝粉尘区域附近1.5 m范围内空气的温度高达1 600 K.模拟程序可用于铝粉尘爆轰的后效研究,对工业安全及其防护具有指导意义.     

Time reversing solitary waves

We present new results for the time reversal of nonlinear pulses traveling in a random medium, in particular for solitary waves. We consider long water waves propagating in the presence of a spatially random depth. Both hyperbolic and dispersive regimes are considered. We demonstrate that in the presence of properly scaled stochastic forcing the solution to the nonlinear (shallow water) conservation law is regularized leading to a viscous shock profile. This enables time-reversal experiments beyond the critical time for shock formation. Furthermore, we present numerical experiments for the time-reversed refocusing of solitary waves in a regime where theory is not yet available. Solitary wave refocusing simulations are performed with a new Boussinesq model, both in transmission and in reflection.     

用STC格式求解二维激波—边界层相互作用问题

将空间—时间守恒(STC)格式应用于求解N-S方程,并对激波—边界层相互作用问题进行了计算。结果表明,该方法可捕获激波与边界层相互作用的各种现象,显示了优良的数值模拟性能。     

Time fractional effect on ion acoustic shock waves in ion-pair plasma

The nonlinear properties of ion acoustic shock waves are studied. The Burgers equation is derived and converted into the time fractional Burgers equation by Agrawal’s method. Using the Adomian decomposition method, shock wave solutions of the time fractional Burgers equation are constructed. The effect of the time fractional parameter on the shock wave properties in ion-pair plasma is investigated. The results obtained may be important in investigating the broadband electrostatic shock noise in D- and F-regions of Earth’s ionosphere.     

Semianalytical solution of the problem of converging shock waves

An approach to the determination of the self-similarity parameter in the problem of converging strong shock waves is suggested. This approach allows one to obtain analytical expressions that approximate the numerical solution. For adiabatic constants gamma = 6/5-7, the values of the obtained self-similarity parameter differ by <1% from the values determined by the numerical procedure. In addition, accurate analytical characteristics of the reflected shock wave are obtained.     

BGK方法在非结构网格上的应用

采用旋转局部坐标的方法,发展了一种针对非结构网格的BGK计算方法.该方法属于有限体积法,大致分为两个步骤:①空间离散:②通量计算及时间推进.在第①步中,采用基于最小二乘法的高阶ENO格式来获得宏观物理量的高阶导数;在第②步中,采用旋转坐标轴的方法来计算非结构网格单元各边的通量.并得出了后台阶绕流(Backward Facing Step)及翼型绕流(Flow Over an Airfoil)两个算例的计算结果.     

Method of adaptive artificial viscosity

A new finite-difference method for the numerical solution of gas dynamics equations is proposed. This method is a uniform monotonous finite-difference scheme of second-order approximation on time and space outside of domains of shock and compression waves. This method is based on inputting adaptive artificial viscosity (AAV) into gas dynamics equations. In this paper, this method is analyzed for 2D geometry. The testing computations of the movement of contact discontinuities and shock waves and the breakup of discontinuities are demonstrated.     

Bifurcation of Nonclassical Viscous Shock Profiles from the Constant State

We determine the bifurcation from the constant solution of nonclassical transitional and overcompressive viscous shock profiles, in regions of strict hyperbolicity. Whereas classical shock waves in systems of conservation laws involve a single characteristic field, nonclassical waves involve two fields in an essential way. This feature is reflected in the viscous profile differential equation, which undergoes codimension-three bifurcation of the kind studied by Dumortier et al., as opposed to the codimension-one bifurcation occurring in the classical case. We carry out a complete bifurcation analysis for systems of two quadratic conservation laws with constant, strictly parabolic viscosity matrices by reducing to a canonical form introduced by Fiddelaers. We show that all such systems, except possibly those on a codimension-one variety in parameter space, give rise to nonclassical shock waves, and we classify the number and types of their bifurcation points. One consequence of our analysis is that weak transitional waves arise in pairs, with profiles forming a 2-cycle configuration previously shown to lead to nonuniqueness of Riemann solutions and to nontrivial asymptotic dynamics of the conservation laws. Another consequence is that appearance of weak nonclassical waves is necessarily associated with change of stability in constant solutions of the parabolic system of conservation laws, rather than with change of type in the associated hyperbolic system.     

非结构网格下非定常流场双时间步长的加权ENO-强紧致杂交高分辨率格式

针对三维非定常、可压缩流场的Navier-Stokes方程组,本文提出一种新的双时间步长高精度快速迭代格式。该格式在时间上具有二阶精度,在空间离散上不低于三阶。在对流项与粘性项的处理上,本格式分别采用了加权ENO-强紧致格式与紧致四阶精度格式的思想。几个典型算例的实践表明:计算结果与相关实验数据比较吻合,初步表明了该算法可以在非结构网格下具有高效率与高分辨率的特征。     

FCT有限元法数值求解三维可压缩高速无粘流问题

从Euler方程出发,利用流量修正有限元法(FEM-FCT)求解三维无粘流动的高速流场。通过对圆球、简化航天飞机和双子星座飞船返回舱外形的数值模拟证明,这种方法对激波有较高的分辨能力,是一种有效的方法。     

CE/SE方法数值模拟炸药粉尘爆轰

采用CE/SE方法数值模拟悬浮在空气中的RDX炸药粉尘的两相爆轰过程.炸药颗粒在爆轰波阵面后的高温高速气流中加速并升温,释放能量支持爆轰波传播.数值模拟爆轰波管中的粉尘爆轰,得到爆轰波流场中的物理量分布,确定爆轰参数,数值结果与文献符合较好.数值模拟复杂通道中的炸药粉尘爆轰,预测了爆轰波的发展和传播过程以及爆轰波后的流场演化.数值结果表明CE/SE方法能成功模拟气体-固体两相爆轰,为粉尘爆轰的研究提供了新的数值预测手段.     

Time-Asymptotic Behavior of Wave Propagation Around a Viscous Shock Profile

We study the nonlinear stability of shock waves for viscous conservation laws. Our approach is based on a new construction of a fundamental solution for a linearized system around a shock profile. We obtain, for the first time, the pointwise estimates of nonlinear wave interactions across a shock wave. Our results apply to all ranges of weak shock waves and small perturbations. In particular, our results reduce to the time-asymptotic behavior of constant state perturbation, uniformly as the strength of the shock wave tends to zero. The research of the first author was partially supported by NSC Grant 96-2628-M-001-011 and NSF Grant DMS-0709248. The research of the second author was partially supported by NSF Grant DMS-0207154 and UAB Advance Program, sponsored by NSF.     

Delta shock waves in conservation laws with impulsive moving source: some results obtained by multiplying distributions

The present paper concerns the study of a Riemann problem for the conservation law u_t + [? (u)]_x = kδ (x ? vt) where x, t, k, v and u = u(x,t) are real numbers. We consider ? an entire function taking real values on the real axis and δ stands for the Dirac measure. Within a convenient space of distributions we will explicitly see the possible emergence of waves with the shape of shock waves, delta waves and delta shock waves. For this purpose, we define a rigorous concept of a solution which extends both the classical solution concept and a weak solution concept. All this framework is developed in the setting of a distributional product that is not constructed by approximation. We include the main ideas of this product for the reader’s convenience. Recall that delta shock waves are relevant physical phenomena which may be interpreted as processes of concentration of mass or even as processes of formation of galaxies in the universe.