Material point method of modelling and simulation of reacting flow of oxygen

Aerospace vehicles are continually being designed to sustain flight at higher speeds and higher altitudes than previously attainable. At hypersonic speeds, gases within a flow begin to chemically react and the fluid's physical properties are modified. It is desirable to model these effects within the Material Point Method (MPM). The MPM is a combined Eulerian–Lagrangian particle-based solver that calculates the physical properties of individual particles and uses a background grid for information storage and exchange. This study introduces chemically reacting flow modelling within the MPM numerical algorithm and illustrates a simple application using the AeroElastic Material Point Method (AEMPM) code. The governing equations of reacting flows are introduced and their direct application within an MPM code is discussed. A flow of 100% oxygen is illustrated and the results are compared with independently developed computational non-equilibrium algorithms. Observed trends agree well with results from an independently developed source.     

Analysis of aerodynamic problems with geometrically unspecified boundaries using an enhanced Lagrangian method

This paper presents solutions for several 2-D aerodynamic problems with geometrically unspecified boundaries. These solutions are obtained with an enhanced Lagrangian method based on stream function and Lagrangian-distance coordinates, which includes special procedures to substantially improve the numerical resolution of the shock waves and for the numerical implementation of the aerodynamic conditions defining the geometrically unspecified solid or fluid boundaries. The method is first validated for flows with specified solid boundaries by comparison with exact analytical solutions and with previous computational results obtained by numerical methods using Eulerian formulations. In all cases, this enhanced Lagrangian method displayed a very good accuracy and computational efficiency and a sharp numerical resolution of shock waves. Then this method is used to obtain solutions for problems with geometrically unspecified boundaries, such as: (i) indirect problems of determining the geometrical shape of airfoils and nozzle walls for a specified pressure distribution; (ii) supersonic nozzle design problem for a specified uniform flow at the nozzle outlet based on reflection-suppression condition; (iii) analysis of flexible-membrane airfoils; and (iv) analysis of jet-flapped airfoils.     

The intersection marker method for 3D interface tracking of deformable surfaces in finite volumes

Currently, the majority of computational fluid dynamics (CFD) codes use the finite volume method to spatially discretise the computational domain, sometimes as an array of cubic control volumes. The Finite volume method works well with single‐phase flow simulations, but two‐phase flow simulations are more challenging because of the need to track the surface interface traversing and deforming within the 3D grid. Surface area and volume fraction details of each interface cell must be accurately accounted for, in order to calculate for the momentum exchange and rates of heat and mass transfer across the interface. To attain a higher accuracy in two‐phase flow CFD calculations, the intersection marker (ISM) method is developed. The ISM method is a hybrid Lagrangian–Eulerian front‐tracking algorithm that can model an arbitrary 3D surface within an array of cubic control volumes. The ISM method has a cell‐by‐cell remeshing capability that is volume conservative and is suitable for the tracking of complex interface deformation in transient two‐phase CFD simulations. Copyright © 2015 John Wiley & Sons, Ltd.     

A displacement-based reference frame formulation for steady-state thermo-elasto-plastic material processes

A reference frame formulation for steady state thermo-elasto-plastic processes is presented. The displacement and history dependent response fields appear as the primary variables in this mixed formulation. Unlike displacement based Lagrangian formulations, our formulation does not require a transient analysis to simulate a steady state process and yields results that are free of numerical oscillations and which require considerably less computational effort. And unlike velocity based Eulerian methods, our formulation does not require free surface corrections or streamline integration algorithms. A laser surface treatment process is simulated and our results are in agreement with those obtained from a computationally intensive transient Lagrangian analysis.     

A time splitting fictitious domain algorithm for fluid–structure interaction problems (A fictitious domain algorithm for FSI)

A Finite Element Method in mixed Eulerian and Lagrangian formulation is developed to allow direct numerical simulations of dynamical interaction between an incompressible fluid and a hyper-elastic incompressible solid. A Fictitious Domain Method is applied so that the fluid is extended inside the deformable solid volume and the velocity field in the entire computational domain is resolved in an Eulerian framework. Solid motion, which is tracked in a Lagrangian framework, is imposed through the body force acting on the fluid within the solid boundaries. Solid stress smoothing on the Lagrangian mesh is performed with the Zienkiewicz–Zhu patch recovery method. High-order Gaussian integration quadratures over cut elements are used in order to avoid sub-meshing within elements in the Eulerian mesh that are intersected by the Lagrangian grid. The algorithm is implemented and verified in two spatial dimensions by comparing with the well validated simulations of solid deformation in a lid driven cavity and periodic elastic wall deformation driven by a time-dependent flow. It shows good agreement with the numerical results reported in the literature. In 3-D the method is validated against previously reported numerical simulations of 3-D rhythmically contracting alveolated ducts.     

Three-dimensional arbitrary Lagrangian–Eulerian numerical prediction method for non-linear free surface oscillation

A numerical prediction method has been proposed to predict non-linear free surface oscillation in an arbitrarily-shaped three-dimensional container. The liquid motions are described with Navier–Stokes equations rather than Laplace equations which are derived by assuming the velocity potential. The profile of a liquid surface is precisely represented with the three-dimensional curvilinear co-ordinates which are regenerated in each computational step on the basis of the arbitrary Lagrangian–Eulerian (ALE) formulation. In the transformed space, the governing equations are discretized on a Lagrangian scheme with sufficient numerical accuracy and the boundary conditions near the liquid surface are implemented in a complete manner. In order to confirm the applicability of the present computational technique, numerical simulations are conducted for the free oscillations of viscid and inviscid liquids and for highly non-linear oscillation. In addition, non-linear sloshing motions caused by horizontal and vertical excitations and a transition from non-linear sloshing to swirling are numerically predicted in three-dimensional cylindrical containers. Conclusively, it is shown that these sloshing motions associated with high non-linearity are reasonably predicted with the present numerical technique. © 1998 John Wiley & Sons, Ltd.     

圆柱形汇聚激波诱导 Richtmyer-Meshkov不稳定的 SPH 模拟

汇聚激波诱导不同物质界面的Richtmyer-Meshkov(RM)不稳定现象在惯性约束核聚变领域有重要的学术意义和工程背景.基于网格离散的宏观流体力学方法由于数值扩散问题往往需要高阶精度算法才能准确追踪界面演化,且对大变形和破碎合并等复杂界面追踪也极为困难.光滑粒子流体动力学(smoothed particlehydrodynamics,SPH)方法采用纯拉格朗日算法,可以有效克服上述难点.但经典SPH算法需采用人工黏性处理强间断,在激波间断处往往会出现严重的非物理振荡,对于涉及强冲击不稳定性问题,很难达到理想的模拟效果.本文采用基于HLL黎曼求解器的SPH算法,实现了对强激波和大密度比物质界面的有效分辨和追踪.一维数值校核证明了代码的可靠性、健壮性,并进一步模拟了二维圆柱形汇聚冲击波冲击四边形轻/重气界面诱导的RM不稳定性问题,与已有实验结果进行了对比,发现模拟结果与实验结果吻合.通过分析界面演化过程中的密度及压力变化,发现本文所采用的方法可准确地追踪激波与界面作用的复杂界面和波系演化规律.研究结果为进一步理解和解释汇聚冲击条件下的RM不稳定性机理奠定了基础.      

基于隐式扩散的直接力格式浸没边界格子Boltzmann方法

采用浸没边界格子Boltzmann (immersed boundary-lattice Boltzmann, IB-LB)模型执行动边界绕流数值模拟时,信息交互界面和边界力计算格式直接影响流动求解器的数值精度和计算效率.基于隐式扩散界面,一种改进的直接力格式IB-LB模型被提出.边界力表达式基于欧拉/拉格朗日变量同一性准则推导,转换矩阵描述的信息交互界面耦合了拉格朗日节点间的非同步运动.采用Richardson迭代数值求解关联边界力与无滑移速度约束的线性方程组,不仅克服了传统速度修正格式中矩阵求逆引起的计算效率问题,而且摆脱了算法稳定性对拉格朗日点分布的依赖.根据解析解已知的Taylor-Green涡流评估本文模型的数值模拟精度,结果表明改进的IB模型能够完整保留背景LB模型的二阶数值精度.静止圆柱和振荡圆柱绕流数值实验结果表明,当前模型在涉及复杂外形和运动界面的流动模拟中能够提供可靠的数值预测,满足力同一性的IB-LB模型能够有效抑制非定常流体力的伪物理震荡.波动翼型绕流模拟验证了当前模型的实用性,可在大变形柔性体流固耦合动力学问题中进一步推广.     

Numerical study of the lift force influence on two-phase shock tube boundary layer characteristics

The importance of the lift force acting on the dispersed phase in the boundary layer of a laminar gas-particle dilute mixture flow generated by a shock wave is investigated numerically. The particle phase is supposed to form a continuum and is described by an Eulerian approach. The ability of the Eulerian model to simulate particle flows and the importance of the two-way coupling are proven by comparison with experimental data as well as with the numerical results from schemes based on a Lagrangian approach. The models used for the lift force are discussed through comparisons between numerical and experimental results found in the literature. Some results about the formation of a dust cloud are numerically reproduced and show the major role of the lift force. Simulations of two-dimensional two-phase shock tube flows are also performed including the lift force effects. Although the wave propagation is weakly influenced by the lift force, the force modifies substantially the dynamics of the flow near the wall. Received 17 February 2000 / Accepted 30 November 2000     

A Free-Lagrange method for unsteady compressible flow: simulation of a confined cylindrical blast wave

A Free-Lagrange numerical procedure for the simulation of two-dimensional inviscid compressible flow is described in detail. The unsteady Euler equations are solved on an unstructured Lagrangian grid based on a density-weighted Voronoi mesh. The flow solver is of the Godunov type, utilising either the HLLE (2 wave) approximate Riemann solver or the more recent HLLC (3 wave) variant, each adapted to the Lagrangian frame. Within each mesh cell, conserved properties are treated as piece-wise linear, and a slope limiter of the MUSCL type is used to give non-oscillatory behaviour with nominal second order accuracy in space. The solver is first order accurate in time. Modifications to the slope limiter to minimise grid and coordinate dependent effects are described. The performances of the HLLE and HLLC solvers are compared for two test problems; a one-dimensional shock tube and a two-dimensional blast wave confined within a rigid cylinder. The blast wave is initiated by impulsive heating of a gas column whose centreline is parallel to, and one half of the cylinder radius from, the axis of the cylinder. For the shock tube problem, both solvers predict shock and expansion waves in good agreement with theory. For the HLLE solver, contact resolution is poor, especially in the blast wave problem. The HLLC solver achieves near-exact contact capture in both problems. Received May 25, 1995 / Accepted September 11, 1995     

Transient 3D flow of polymer solutions: A Lagrangian computational method

We describe a computational method for the numerical simulation of three-dimensional transient flows of polymer solutions that extends the work of Harlen et al. [O.G. Harlen, J.M. Rallison, P. Szabó, A split Lagrangian–Eulerian method for simulating transient viscoelastic flows, J. Non-Newtonian Fluid Mech. 60 (1995) 81–104]. The method uses a Lagrangian computation of the stress together with an Eulerian computation of the velocity field. Adaptive mesh reconnection based on Delaunay tetrahedra is used to ensure well-shaped elements. Additional shape-quality improvement procedures are developed to improve the algorithm. We validate the method for the benchmark problem of a rigid sphere falling in a cylindrical pipe. Inertia is neglected. We compare results for the axisymmetric case with previous work (using a FENE model), and then consider the off-axis non-axisymmetric case. In the latter case, we find that as the sphere falls, it drifts across the pipe, a phenomenon previously observed in experiments but not fully explained. The physical mechanisms that cause the time-dependent drift are identified, and a simple model based on the normal stresses in the fluid is shown to predict the magnitude of the drift velocity.We also consider a second benchmark problem involving a constriction in an axisymmetric pipe. Numerical difficulties associated with ill-shaped elements near the concave boundary arise for higher Weissenberg numbers. The merits and drawbacks of the new numerical method, and its applicability to various flow geometries are discussed.     

跨音速极限环型颤振的高效数值分析方法

事先建立一个低阶的非线性、非定常气动力模型是开展非线性流场中气动弹性问题研究的一个捷径. 基于CFD方法, 通过计算结构在流场中自激振动的响应来获得系统的训练数据. 采用带输出反馈的循环RBF神经网络, 建立时域非线性气动力降阶模型.耦合结构运动方程和非线性气动力降阶模型, 采用杂交的线性多步方法计算结构在不同速度(动压)下的响应历程, 从而获得模型极限环随速度(动压)变化的特性. 两个典型的跨音速极限环型颤振算例表明, 基于气动力降阶模型方法的计算结果与直接CFD仿真结果吻合很好, 与后者相比其将计算效率提高了1~2个数量级.      

The characteristic‐based split (CBS) meshfree method for free surface flow problems in ALE formulation

A semi‐implicit characteristic‐based split (CBS) meshfree algorithm in the arbitrary Lagrangian Eulerian (ALE) framework is proposed for the numerical solution of incompressible free surface flow problem in the paper. The algorithm is the extension of general CBS method which was initially introduced in finite element framework, this is due to the fact that CBS method not only can enhance the stability, but also avoid LBB condition when equal order basis function is used to approximate velocity and pressure variables. Meanwhile, a simple way for node update and node speed calculation is developed which is used to capture the free surface exactly. The numerical solutions are compared with available analytical and numerical solutions, which shows that the proposed method has better ability to simulate the free surface incompressible flow problem. Copyright © 2009 John Wiley & Sons, Ltd.     

On some optimization procedures for shock localization

We consider a problem on shock wave localization in the numerical solution of one-dimensional unsteady problems of gas dynamics in Eulerian variables obtained on the basis of finite difference shock-capturing schemes. An optimization method for strong discontinuity localization proposed previously by Miranker and Pironneau is investigated by means of methods of classical variational calculus. This method may be difficult to implement when the entropy condition is included in the formulation of Miranker and Pironneau's optimization problem as an active constraint. In this connection we suggest an alternative optimization problem using artificial viscosity in the variational principle. It is shown theoretically that the application of such a variational principle yields a trajectory which coincides with the true discontinuity trajectory in the case of a shock wave moving at a constant speed. On the basis of this modification one more algorithm is proposed which reduces the shock localization problem to a problem of minimization of a univariate function. Numerical tests corroborate completely the theoretical conclusions. In particular, a higher shock localization accuracy is obtained on the basis of the proposed algorithms as compared to the original Miranker-Pironneau method.     

Interfacial flow computations using adaptive Eulerian–Lagrangian method for spacecraft applications

Understanding the interfacial dynamics and fluid physics associated with the operation of spacecraft is important for scientific as well as engineering purposes. To help address the issues associated with moving boundaries, interfacial dynamics, and spatial‐temporal variations in time and length scales, a 3‐D adaptive Eulerian–Lagrangian method is extended and further developed. The stationary (Eulerian) Cartesian grid is adopted to resolve the fluid flow, and the marker‐based triangulated moving (Lagrangian) surface meshes are utilized to treat the phase boundary. The key concepts and numerical procedures, along with the selected interfacial flow problems are presented. Specifically, the liquid fuel draining dynamics in different flow regimes, and the liquid surface stability under vertically oscillating gravitational acceleration are investigated. Direct assessment of experimental measurement and scaling analysis is made to highlight the computational performance of the present approach as well as the key fluid physics influenced by the given flow parameters. Copyright © 2010 John Wiley & Sons, Ltd.     

Comparisons of some difference forms for compressible flow in cylindrical geometry on arbitrary Lagrangian and Eulerian framework

The study of cylindrically symmetric compressible fluid is interesting from both theoretical and numerical points of view. In this paper, the typical spherical symmetry properties of the numerical schemes are discussed, and an area weighted scheme is extended from a Lagrangian method to an arbitrary Lagrangian and Eulerian (ALE) method. Numerical results are presented to compare three discrete configurations, i.e., the control volume scheme, the area weighted scheme, and the plane scheme with the addition of a geometrical source. The fact that the singularity arises from the geometrical source term in the plane scheme is illustrated. A suggestion for choosing the discrete formulation is given when the strong shock wave problems are simulated.     

Theoretical analysis for achieving high‐order spatial accuracy in Lagrangian/Eulerian source terms

In a fully coupled Lagrangian/Eulerian two‐phase calculation, the source terms from computational particles must be agglomerated to nearby gas‐phase nodes. Existing methods are capable of accomplishing this particle‐to‐gas coupling with second‐order accuracy. However, higher‐order methods would be useful for applications such as two‐phase direct numerical simulation and large eddy simulation. A theoretical basis is provided for producing high spatial accuracy in particle‐to‐gas source terms with low computational cost. The present work derives fourth‐ and sixth‐order accurate methods, and the procedure for even higher accuracy is discussed. The theory is also expanded to include two‐ and three‐dimensional calculations. One‐ and two‐dimensional tests are used to demonstrate the convergence of this method and to highlight problems with statistical noise. Finally, the potential for application in computational fluid dynamics codes is discussed. It is concluded that high‐order kernels have practical benefits only under limited ranges of statistical and spatial resolution. Additionally, convergence demonstrations with full CFD codes will be extremely difficult due to the worsening of statistical errors with increasing mesh resolution. Copyright © 2006 John Wiley & Sons, Ltd.     

Multiscale simulations and experiments on water jet atomization

Numerical simulation of primary atomization at high Reynolds number is still a challenging problem. In this work a multiscale approach for the numerical simulation of liquid jet primary atomization is applied, using an Eulerian-Lagrangian coupling. In this approach, an Eulerian volume of fluid (VOF) method, where the Reynolds stresses are closed by a Reynolds stress model is applied to model the global spreading of the liquid jet. The formation of the micro-scale droplets, which are usually smaller than the grid spacing in the computational domain, is modelled by an energy-based sub-grid model. Where the disruptive forces (turbulence and surface pressure) of turbulent eddies near the surface of the jet overcome the capillary forces, droplets are released with the local properties of the corresponding eddies. The dynamics of the generated droplets are modelled using Lagrangian particle tracking (LPT). A numerical coupling between the Eulerian and Lagrangian frames is then established via source terms in conservation equations. As a follow-up study to our investigation in Saeedipour et al. (2016a), the present paper aims at modelling drop formation from liquid jets at high Reynolds numbers in the atomization regime and validating the simulation results against in-house experiments. For this purpose, phase-Doppler anemometry (PDA) was used to measure the droplet size and velocity distributions in sprays produced by water jet breakup at different Reynolds numbers in the atomization regime. The spray properties, such as droplet size spectra, local and global Sauter-mean drop sizes and velocity distributions obtained from the simulations are compared with experiment at various locations with very good agreement.