A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

固体炸药爆轰的一种考虑热学非平衡的反应流动模型

基于固体炸药爆轰过程中化学反应混合区内的固相反应物与气相生成物处于力学平衡状态及热学非平衡状态的事实,提出一种考虑热学非平衡效应的反应流动模型来描述固体炸药的爆轰流动现象.该爆轰流动模型的主要特点是,在反应混合物Euler方程和固相反应物质量守恒方程的基础上,通过附加一套关于固相反应物的组分物理量的流动控制方程来表达固相反应物与气相生成物之间的热学非平衡效应.根据反应混合区内固相反应物与气相生成物这两种化学组分保持各自内能守恒的混合规则,并借助它们具有压力相等的性质以及满足体积分数总和为1的条件,推导获得的附加方程有：固相反应物的内能演化方程、体积分数演化方程及反应混合物的压力演化方程.这样,建立的爆轰模型包括：反应混合物的质量守恒方程、动量守恒方程、总能量守恒方程、压力演化方程,以及固相反应物的质量守恒方程、内能演化方程、体积分数演化方程.对所获得的爆轰模型方程组采用一个时空二阶精度的有限体积法进行数值求解,典型爆轰问题算例结果表明本文提出的固体炸药爆轰模型是合理的.     

多组份流动质量分数保极值原理算法

对基于质量分数的Mie-Gruneisen状态方程多流体组份模型提出了新的数值方法.该模型保持混合流体的质量、动量、和能量守恒,保持各组份分质量守恒,在多流体组份界面处保持压力和速度一致.该模型是拟守恒型方程系统.对该模型系统的离散采用波传播算法.与直接对模型中所有守恒方程采用相同算法不同的是,在处理分介质质量守恒方程时,对波传播算法进行了修正,使之满足质量分数保极值原理.而不作修改的算法则不能保证质量分数在[0,1]范围.数值实验验证了该方法有效.     

FIVER: A finite volume method based on exact two-phase Riemann problems and sparse grids for multi-material flows with large density jumps

A robust finite volume method for the solution of high-speed compressible flows in multi-material domains involving arbitrary equations of state and large density jumps is presented. The global domain of interest can include a moving or deformable subdomain that furthermore may undergo topological changes due to, for example, crack propagation. The key components of the proposed method include: (a) the definition of a discrete surrogate material interface, (b) the computation of a reliable approximation of the fluid state vector on each side of a discrete material interface via the construction and solution of a local, exact, two-phase Riemann problem, (c) the algebraic solution of this auxiliary problem when the equation of state allows it, and (d) the solution of this two-phase Riemann problem using sparse grid tabulations otherwise. The proposed computational method is illustrated with the three-dimensional simulation of the dynamics of an underwater explosion bubble.     

蒸汽-冷流体接触冷凝流动的数值模拟

介绍了关于蒸汽-冷流体直接接触冷凝流动与传热的数值计算模型与部分研究结果。用Level Set方法确定蒸汽-冷流体接触界面的位置和形状,建立了对蒸汽和冷流体普遍适用的动量、能量和质量守恒方程,在能量和质量寺恒方程中增加了部分项用于计算蒸汽冷凝所产生的影响。用有限差分法在交错网格上离散控制方程,用Runge-Kutta法-五阶WENO组合格式求解Level Set输运方程,用压力修正的迭代Projection方法求解动量方程,而用SIMPLE方法求解温度控制方程。对算例的计算结果表明,本文所建立的数值计算模型能反映物理现象的宏观特性。根据计算结果,分析了本文模型的优缺点,并指出了今后改进的方向。     

Fully implicit 1D radiation hydrodynamics: Validation and verification

A fully implicit finite difference scheme has been developed to solve the hydrodynamic equations coupled with radiation transport. Solution of the time-dependent radiation transport equation is obtained using the discrete ordinates method and the energy flow into the Lagrangian meshes as a result of radiation interaction is fully accounted for. A tridiagonal matrix system is solved at each time step to determine the hydrodynamic variables implicitly. The results obtained from this fully implicit radiation hydrodynamics code in the planar geometry agrees well with the scaling law for radiation driven strong shock propagation in aluminium. For the point explosion problem the self similar solutions are compared with results for pure hydrodynamic case in spherical geometry. Results obtained when radiation interaction is also accounted agree with those of point explosion with heat conduction for lower input energies. Having, thus, benchmarked the code, self convergence of the method w.r.t. time step is studied in detail for both the planar and spherical problems. Spatial as well as temporal convergence rates are ?1 as expected from the difference forms of mass, momentum and energy conservation equations. This shows that the asymptotic convergence rate of the code is realized properly.     

A conservative level set method suitable for variable-order approximations and unstructured meshes

This paper presents a formulation for free-surface computations capable of handling complex phenomena, such as wave breaking, without excessive mass loss or smearing of the interface. The formulation is suitable for discretizations using finite elements of any topology and order, or other approaches such as isogeometric and finite volume methods. Furthermore, the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties. Implementations of the method are tested on several difficult two- and three-dimensional problems, including two incompressible air/water flow problems with available experimental results. Linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested. Global conservation and agreement with experiments as well as computations by other researchers are obtained.     

Stabilization of the Eulerian model for incompressible multiphase flow by artificial diffusion

The commonly used Eulerian or continuum model for incompressible multiphase flow is known to be unstable to perturbations for all wavenumbers, even if viscosity terms are used in the momentum equations. In the present work the model is stabilized by adding explicit artificial diffusion to the mass equations. The artificial diffusion terms lead to improved stability properties: uniform flow becomes linearly stable for large wavenumbers, and above an analytically derived threshold for the artificial diffusivity, stability for all wavenumbers is achieved. The artificial diffusivity reappears in the momentum equations, in such a way that fundamental properties of the standard equations remain valid: Galilean invariance is maintained, total mass and momentum are conserved, decay of total kinetic energy is ensured in the absence of external forces, and a flow initially at rest at hydrostatic pressure remains unchanged, even if the spatial distribution of volume fractions is nonuniform. A staggered finite volume pressure correction method using central differencing (leading to energy conserving discretization of convective and pressure terms) is presented. Application of the method to one-dimensional two-phase flow of falling particles particles confirms that the equations are stable with and unstable without artificial diffusion in the volume fraction equation.     

Modeling of multicomponent-fuel drop-laden mixing layers having a multitude of species

A formulation representing multicomponent-fuel (MC-fuel) composition as a probability distribution function (PDF) depending on the molar mass is used to construct a model of a large number of MC-fuel drops evaporating in a gas flow, so as to assess the extent of fuel specificity on the vapor composition. The PDF is a combination of two Gamma PDFs, which was previously shown to duplicate the behavior of a fuel composed of many species during single drop evaporation. The conservation equations are Eulerian for the flow and Lagrangian for the physical drops, all of which are individually followed. The gas conservation equations for mass, momentum, species, and energy are complemented by differential conservation equations for the first four moments of the gas-composition PDF; all coupled to the perfect gas equation of state. Source terms in all conservation equations couple the gas phase to the drops. The drop conservation equations for mass, position, momentum, and energy are complemented by differential equations for four moments of the liquid-composition PDF. The simulations are performed for a three-dimensional mixing layer whose lower stream is initially laden with drops. Initial perturbations excite the layer to promote the double pairing of its four initial spanwise vortices to an ultimate vortex. The drop temperature is initially lower than that of the surrounding gas, initiating drop heating and evaporation. The results focus on both evolution and the state of the drops and gas when layers reach a momentum-thickness maximum past the double vortex pairing; particular emphasis is on the gas composition. Comparisons between simulations with n-decane, diesel, and three kerosenes show that at same initial Reynolds number and Stokes number distribution, a single-component fuel cannot represent MC fuels. Substantial differences among the MC-fuel vapor composition indicate that fuel specificity must be captured for the prediction of combustion.     

Compact fourth-order finite volume method for numerical solutions of Navier–Stokes equations on staggered grids

The development of a compact fourth-order finite volume method for solutions of the Navier–Stokes equations on staggered grids is presented. A special attention is given to the conservation laws on momentum control volumes. A higher-order divergence-free interpolation for convective velocities is developed which ensures a perfect conservation of mass and momentum on momentum control volumes. Three forms of the nonlinear correction for staggered grids are proposed and studied. The accuracy of each approximation is assessed comparatively in Fourier space. The importance of higher-order approximations of pressure is discussed and numerically demonstrated. Fourth-order accuracy of the complete scheme is illustrated by the doubly-periodic shear layer and the instability of plane-channel flow. The efficiency of the scheme is demonstrated by a grid dependency study of turbulent channel flows by means of direct numerical simulations. The proposed scheme is highly accurate and efficient. At the same level of accuracy, the fourth-order scheme can be ten times faster than the second-order counterpart. This gain in efficiency can be spent on a higher resolution for more accurate solutions at a lower cost.     

Viscous quark‐gluon plasma in the early universe

In the present work a study is given for the evolution of a flat, isotropic and homogeneous Universe, which is filled with a causal bulk viscous cosmological fluid. We describe the viscous properties by an ultra‐relativistic equation of state, and bulk viscosity coefficient obtained from recent lattice QCD calculations. The basic equation for the Hubble parameter is derived by using the energy equation obtained from the assumption of the covariant conservation of the energy‐momentum tensor of the matter in the Universe. By assuming a power law dependence of the bulk viscosity coefficient, temperature and relaxation time on the energy density, we derive the evolution equation for the Hubble function. By using the equations of state from recent lattice QCD simulations and heavy‐ion collisions we obtain an approximate solution of the field equations. In this treatment for the viscous cosmology, no evidence for singularity is observed. For example, both the Hubble parameter and the scale factor are finite at t = 0, where t is the comoving time. Furthermore, their time evolution essentially differs from the one associated with non‐viscous and ideal gas. Also it is noticed that the thermodynamic quantities, like temperature, energy density and bulk pressure remain finite. Particular solutions are also considered in order to prove that the free parameter in this model does qualitatively influence the final results.     

声子气的状态方程和声子气运动的守恒方程

根据爱因斯坦的狭义相对论中质能的等效关系，把固体(本文指非导体)晶格(原子)的质量分为晶格(原子)的静质量和晶格热振动能量的等效质量两个部分，后者就是固体中声子气的等效质量.晶格(原子)热振动的能量则分为晶格(原子)静质量具有的热能以及声子气质量具有的热能.基于固体的状态方程，导得了晶格静质量热振动的状态方程和声子气的状态方程.声子气在固体介质中的宏观运动就是热量在固体中的传递过程.建立了声子气运动的守恒方程组，分析表明，忽略惯性力时声子气的动量守恒方程就退化为傅里叶导热定律，阐明了傅里叶导热定律的物理本质是声子气驱动力与阻力的平衡方程.当热流密度很大惯性力不能忽略时，傅里叶导热定律不再适用. 关键词： 非傅里叶导热 声子气 声子气质量 状态方程 守恒方程     

A conservative discrete compatibility-constraint low-Mach pressure-correction algorithm for time-accurate simulations of variable density flows

In this paper, we develop the discrete compatibility-constraint pressure-correction algorithm for transient simulations of variable density flows at low-Mach numbers. The constraint for the velocity field is constructed from a combination of the discrete equations of continuity and scalar (e.g. energy) transport, imposing that the newly predicted state must be compatible, in agreement with the equation of state. This way, mass and scalar conservation are guaranteed and the equation of state is exactly fulfilled at every time step. For comparison reasons, two other types of well-known pressure-correction algorithms are also used. The first class, denoted as continuity-constraint pressure-correction, is based on a constraint for the velocity field that is derived solely from the continuity equation. The second class, denoted as analytical compatibility-constraint pressure-correction, constructs the constraint from an analytical combination of the material derivative of the equation of state and the continuity and scalar equations. The algorithms are tested for three example fluid configurations: a single-fluid ideal gas, a two-fluid inert mixture and a two-fluid reacting mixture. The latter is special in the sense that the equation of state is non-linear and not everywhere differentiable. The continuity-constraint pressure-correction algorithm yields unstable solutions if density ratios are high. The analytical compatibility-constraint pressure-correction algorithm yields stable results, but the predicted states do not correspond to the equation of state. The discrete compatibility-constraint pressure-correction algorithm performs well on all test cases: the simulation results are stable and exactly match the equation of state.     

Numerical method satisfying the first two conservation laws for the Korteweg–de Vries equation

In this paper, we develop a finite-volume scheme for the KdV equation which conserves both the momentum and energy. The main ingredient of the method is a numerical device we developed in recent years that enables us to construct numerical method for a PDE that also simulates its related equations. In the method, numerical approximations to both the momentum and energy are conservatively computed. The operator splitting approach is adopted in constructing the method in which the conservation and dispersion parts of the equation are alternatively solved; our numerical device is applied in solving the conservation part of the equation. The feasibility and stability of the method is discussed, which involves an important property of the method, the so-called Jensen condition. The truncation error of the method is analyzed, which shows that the method is second-order accurate. Finally, several numerical examples, including the Zabusky–Kruskal’s example, are presented to show the good stability property of the method for long-time numerical integration.     

轴对称钝体后湍流扩散燃烧的PDF模拟

在无结构网格中，对轴对称钝体驻定的湍流扩散火焰进行数值模拟．采用有限容积(FV)／Monte Carlo(MC)混合算法求解湍流燃烧问题的混合算法．Monte Carlo法求解脉动速度一标量．频率的联合概率密度函数方程，有限容积法求解平均质量、动量和能量方程．求解的两组方程是相容的，合理的耦合方式可以减少统计偏差，计算精度和效率显著优于单独的颗粒方法．文中对化学反应采用层流火焰面模型，并将数值计算结果与实验结果作了比较和分析．     

On the spectral and conservation properties of nonlinear discretization operators

Following the study of Pirozzoli [1], the objective of the present work is to provide a detailed theoretical analysis of the spectral properties and the conservation properties of nonlinear finite difference discretizations. First, a Nonlinear Spectral Analysis (NSA) is proposed in order to study the statistical behavior of the modified wavenumber of a nonlinear finite difference operator, for a large set of synthetic scalar fields with prescribed energy spectrum and random phase. Second, the necessary conditions for local and global conservation of momentum and kinetic energy are derived and verified for nonlinear discretizations. Because the nonlinear mechanisms result in a violation of the energy conservation conditions, the NSA is used to quantify the energy imbalance. Third, the effect of aliasing errors due to the nonlinearity is analyzed. Finally, the theoretical observations are verified for two simple, thought relevant, numerical simulations.     

无结构网格有限体积方法及在热对流中的应用研究

首次将无结构三角网格的有限体积方法和压强连接半隐式算法相结合,用于求解非平行壁管道中的热对流问题。并由此分析了化学汽相淀积薄膜生长的均匀性问题。计算结果对于分析一类管道中热和动量输运现象均有普遍指导意义。     

Geometric discrete analogues of tangent bundles and constrained Lagrangian systems

Discretizing variational principles, as opposed to discretizing differential equations, leads to discrete-time analogues of mechanics, and, systematically, to geometric numerical integrators. The phase space of such variational discretizations is often the set of configuration pairs, analogously corresponding to initial and terminal points of a tangent vector. We develop alternative discrete analogues of tangent bundles, by extending tangent vectors to finite curve segments, one curve segment for each tangent vector. Towards flexible, high order numerical integrators, we use these discrete tangent bundles as phase spaces for discretizations of the variational principles of Lagrangian systems, up to the generality of nonholonomic mechanical systems with nonlinear constraints. We obtain a self-contained and transparent development, where regularity, equations of motion, symmetry and momentum, and structure preservation, all have natural expressions.     

A sweeping preconditioner for time-harmonic Maxwell’s equations with finite elements

This paper is concerned with preconditioning the stiffness matrix resulting from finite element discretizations of Maxwell’s equations in the high frequency regime. The moving PML sweeping preconditioner, first introduced for the Helmholtz equation on a Cartesian finite difference grid, is generalized to an unstructured mesh with finite elements. The method dramatically reduces the number of GMRES iterations necessary for convergence, resulting in an almost linear complexity solver. Numerical examples including electromagnetic cloaking simulations are presented to demonstrate the efficiency of the proposed method.     

热质的运动和传递-热子气的守恒方程和傅立叶定律

本文在热质和热子气概念的基础上建立了热子气的质量、动量、能量守恒方程.基于傅立叶导热定律求得了热子气粘性系数和粘性力的近似表达式.分析了傅立叶定律的物理意义:傅立叶定律是在忽略惯性力的条件下对热子气动量方程的近似.在极低温度或极高热流密度条件下傅立叶定律不再适用.