A particle–gridless hybrid method for incompressible flows

A particle–gridless hybrid method for the analysis of incompressible flows is presented. The numerical scheme consists of Lagrangian and Eulerian phases as in an arbitrary Lagrangian–Eulerian (ALE) method, where a new‐time physical property at an arbitrary position is determined by introducing an artificial velocity. For the Lagrangian calculation, the moving‐particle semi‐implicit (MPS) method is used. Diffusion and pressure gradient terms of the Navier–Stokes equation are calculated using the particle interaction models of the MPS method. As an incompressible condition, divergence of velocity is used while the particle number density is kept constant in the MPS method. For the Eulerian calculation, an accurate and stable convection scheme is developed. This convection scheme is based on a flow directional local grid so that it can be applied to multi‐dimensional convection problems easily. A two‐dimensional pure convection problem is calculated and a more accurate and stable solution is obtained compared with other schemes. The particle–gridless hybrid method is applied to the analysis of sloshing problems. The amplitude and period of sloshing are predicted accurately by the present method. The range of the occurrence of self‐induced sloshing predicted by the present method shows good agreement with the experimental data. Calculations have succeeded even for the higher injection velocity range, where the grid method fails to simulate. Copyright © 1999 John Wiley & Sons, Ltd.     

Provably stable and time‐accurate extensions of Runge–Kutta schemes for CFD computations on moving grids

Extending fixed‐grid time integration schemes for unsteady CFD applications to moving grids, while formally preserving their numerical stability and time accuracy properties, is a nontrivial task. A general computational framework for constructing stability‐preserving ALE extensions of Eulerian multistep time integration schemes can be found in the literature. A complementary framework for designing accuracy‐preserving ALE extensions of such schemes is also available. However, the application of neither of these two computational frameworks to a multistage method such as a Runge–Kutta (RK) scheme is straightforward. Yet, the RK methods are an important family of explicit and implicit schemes for the approximation of solutions of ordinary differential equations in general and a popular one in CFD applications. This paper presents a methodology for filling this gap. It also applies it to the design of ALE extensions of fixed‐grid explicit and implicit second‐order time‐accurate RK (RK2) methods. To this end, it presents the discrete geometric conservation law associated with ALE RK2 schemes and a method for enforcing it. It also proves, in the context of the nonlinear scalar conservation law, that satisfying this discrete geometric conservation law is a necessary and sufficient condition for a proposed ALE extension of an RK2 scheme to preserve on moving grids the nonlinear stability properties of its fixed‐grid counterpart. All theoretical findings reported in this paper are illustrated with the ALE solution of inviscid and viscous unsteady, nonlinear flow problems associated with vibrations of the AGARD Wing 445.6. Copyright © 2011 John Wiley & Sons, Ltd.     

The Lagrangian approach of advective term treatment and its application to the solution of Navier—Stokes equations

A one-dimensional transport test applied to some conventional advective Eulerian schemes shows that linear stability analyses do not guarantee the actual performances of these schemes. When adopting the Lagrangian approach, the main problem raised in the numerical treatment of advective terms is a problem of interpolation or restitution of the transported function shape from discrete data. Several interpolation methods are tested. Some of them give excellent results and these methods are then extended to multi-dimensional cases. The Lagrangian formulation of the advection term permits an easy solution to the Navier-Stokes equations in primitive variables V, p, by a finite difference scheme, explicit in advection and implicit in diffusion. As an illustration steady state laminar flow behind a sudden enlargement is analysed using an upwind differencing scheme and a Lagrangian scheme. The importance of the choice of the advective scheme in computer programs for industrial application is clearly apparent in this example.     

Interface reconstruction with least‐square fit and split Eulerian–Lagrangian advection

Two new volume‐of‐fluid (VOF) reconstruction algorithms, which are based on a least‐square fit technique, are presented. Their performance is tested for several standard shapes and is compared to a few other VOF/PLIC reconstruction techniques, showing in general a better convergence rate. The geometric nature of Lagrangian and Eulerian split advection algorithms is investigated in detail and a new mixed split Eulerian implicit–Lagrangian explicit (EI–LE) scheme is presented. This method conserves the mass to machine error, performs better than split Eulerian and Lagrangian algorithms, and it is only slightly worse than unsplit schemes. However, the combination of the interface reconstruction with the least‐square fit and its advection with the EI–LE scheme appears superior to other existing approaches. Copyright © 2003 John Wiley & Sons, Ltd.     

Tracking accuracy of a semi‐Lagrangian method for advection–dispersion modelling in rivers

There is an increasing need to improve the computational efficiency of river water quality models because: (1) Monte‐Carlo‐type multi‐simulation methods, that return solutions with statistical distributions or confidence intervals, are becoming the norm, and (2) the systems modelled are increasingly large and complex. So far, most models are based on Eulerian numerical schemes for advection, but these do not meet the requirement of efficiency, being restricted to Courant numbers below unity. The alternative of using semi‐Lagrangian methods, consisting of modelling advection by the method of characteristics, is free from any inherent Courant number restriction. However, it is subject to errors of tracking that result in potential phase errors in the solutions. The aim of this article is primarily to understand and estimate these tracking errors, assuming the use of a cell‐based backward method of characteristics, and considering conditions that would prevail in practical applications in rivers. This is achieved separately for non‐uniform flows and unsteady flows, either via theoretical considerations or using numerical experiments. The main conclusion is that, tracking errors are expected to be negligible in practical applications in both unsteady flows and non‐uniform flows. Also, a very significant computational time saving compared to Eulerian schemes is achievable. Copyright © 2006 John Wiley & Sons, Ltd.     

A finite volume scheme for solving anisotropic diffusion on ALE‐AMR grids

In the context of High Energy Density Physics and more precisely in the field of laser plasma interaction, Lagrangian schemes are commonly used. The lack of robustness due to strong grid deformations requires the regularization of the mesh through the use of Arbitrary Lagrangian Eulerian methods. Theses methods usually add some diffusion and a loss of precision is observed. We propose to use Adaptive Mesh Refinement (AMR) techniques to reduce this loss of accuracy. This work focuses on the resolution of the anisotropic diffusion operator on Arbitrary Lagrangian Eulerian‐AMR grids. In this paper, we describe a second‐order accurate cell‐centered finite volume method for solving anisotropic diffusion on AMR type grids. The scheme described here is based on local flux approximation which can be derived through the use of a finite difference approximation, leading to the CCLADNS scheme. We present here the 2D and 3D extension of the CCLADNS scheme to AMR meshes. Copyright © 2017 John Wiley & Sons, Ltd.     

网格与高精度差分计算问题

研究NS方程差分求解时来流雷诺数、计算格式精度和计算网格之间的关系．给出了判定空间三个方向上的粘性贡献在给定雷诺数、格式精度和网格下是否能够正确计入的估计方法．指出在NS方程的二阶差分方法的数值模拟中,由于物面法向采用了压缩网格技术,物面附近的网格间距很小,该方向上的粘性贡献可被计入．但是如果流向和周向的网格较粗,相应的差分方程中的粘性贡献可能落入截断误差相同的量级,因此在精度上等于仍是求解略去流向和周向粘性项的薄层近似方程．指出,高阶精度的差分计算格式,可以避免对网格要求苛刻的困难．并进一步讨论了建立高阶精度格式的问题,提出了建立高阶精度格式应该满足的原则：耗散控制原则以及色散控制原则．为了避免激波附近可能出现的微小非物理振荡,建议发展混合高阶精度格式,即在激波区,采用网格自适应的NND格式,在激波以外的区域,采用按上述原则发展的高阶格式．     

Particle trajectory calculations with a two‐step three‐time level semi‐Lagrangian scheme well suited for curved flows

This study proposes a new two‐step three‐time level semi‐Lagrangian scheme for calculation of particle trajectories. The scheme is intended to yield accurate determination of the particle departure position, particularly in the presence of significant flow curvature. Experiments were performed both for linear and non‐linear idealized advection problems, with different flow curvatures. Results for simulations with the proposed scheme, and with three other semi‐Lagrangian schemes, and with an Eulerian method are presented. In the linear advection problem the two‐step three‐time level scheme produced smaller root mean square errors and more accurate replication of the angular displacement of a Gaussian hill than the other schemes. In the non‐linear advection experiments the proposed scheme produced, in general, equal or better conservation of domain‐averaged quantities than the other semi‐Lagrangian schemes, especially at large Courant numbers. In idealized frontogenesis simulations the scheme performed equally or better than the other schemes in the representation of sharp gradients in a scalar field. The two‐step three‐time level scheme has some computational overhead as compared with the other three semi‐Lagrangian schemes. Nevertheless, the additional computational effort was shown to be worthwhile, due to the accuracy obtained by the scheme in the experiments with large time steps. The most remarkable feature of the scheme is its robustness, since it performs well both for small and large Courant numbers, in the presence of weak as well strong flow curvatures. Copyright © 2009 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.     

Fourth-order quasi-harmonic equations of state for solids of cubic and tetragonal symmetry

General expressions are given for the dependence of the pressure and the effective elastic moduli on deformation and temperature in the form of a Taylor series expansion with respect to elastic and thermal strains. The temperature dependence of these expressions is derived within the quasi-harmonic approximation of lattice dynamics. The expressions are developed in terms of the Lagrangian strain and an alternative strain measure identical with the Eulerian strain for a pure deformation. They are then used to obtain the third- and fourth-order equations of state for crystals of cubic and tetragonal symmetry and to relate the parameters entering these equations to quantities which are commonly (or may be potentially) measured experimentally. It is shown that available ultrasonic data are not completely sufficient to evaluate the parameters of fourth-order equations of state. For tetragonal symmetry, this problem is still in abeyance; while in the cubic case, it is possible to estimate the fourth-order parameters from shock-wave data and so to give illustrative numerical applications of our equations. Finally, the third- and fourth-order Hugoniots and isotherms of Cu and Ag are calculated in terms of both the Lagrangian and Eulerian strain measures.     

NUMERICAL STABILITY ANALYSIS FOR FREE SURFACE FLOWS

A systematic methodology of numerical stability is presented here in the study of numerical properties of mixed Eulerian– Lagrangian schemes for the numerical simulation of non-linear free surface flows. Two different numerical schemes, i.e. a source–doublet panel method and a desingularized method, are investigated. The present work provides theoretical foundations and applications for numerical stability analysis theory. The matrix stability method has been developed to obtain the spectral radii and normal modes associated with free surface discretization. Some examples considered illustrate the usefulness of this analysis. © 1997 by John Wiley & Sons, Ltd. Int. j. numer. methods fluids 24: 893–912, 1997.     

On the efficiency and quality of numerical solutions in CFD problems using the interface strip preconditioner for domain decomposition methods

In this paper, some pathologies found for simple tests solved by means of preconditioned full iterative schemes are presented. According to these results (Sections 4 and 5), the accuracy deterioration observed should be considered as a warning for the final application given to these solutions. Even though it is well known that full iterative solvers are not the best selection for comparison, they were chosen because they are widely used by the computational fluid dynamic (CFD) community for a diversity of complex fluid dynamics applications. FEM simulated solutions are compared with analytical solutions or measured data for problems that have been considered as ‘benchmarks’ in the CFD literature. For this purpose, the study of the solution obtained via parallelized iterative methods that have been extensively used (e.g. conjugate gradients (CG), GMRes global iteration and its variants, ‘overlapping’ and ‘non‐overlapping’ additive Schwarz domain decomposition schemes) in CFD computations and those obtained with the new interface strip preconditioner (J. Comput. Meth. Sci. Engng 2003; Int. J. Numer. Meth. Engng 2005; 62 (13):1873–1894) for the Schur complement method is carried out. The idea is to present the new solver as an alternative to obtain more accurate and faster solutions in the context of monolithic and non‐monolithic schemes applied to a internal/external viscous compressible/incompressible flows around bodies of complex shapes. Therefore, the target of this work is to show how the reliability of CFD codes is affected by the solver selection and why domain decomposition methods should be viewed not only as a more efficient strategy, but also to guarantee the solution quality. Copyright © 2006 John Wiley & Sons, Ltd.     

Numerical aspects of calculation of confined swirling flows with internal recirculation

Predictions were performed for two different confined swirling flows with internal recirculation zones. The convection terms in the elliptic governing equations were discretized using three different finite differencing schemes: hybrid, quadratic upwind interpolation and skew upwind differencing. For each flow case, calculations were carried out with these schemes and successively refined grids were employed. For the turbulent flow case the k-ε turbulence model was used. The predicted cases were a laminar swirling flow investigated by Bornstein and Escudier, and a turbulent low-swirl case studied by Roback and Johnson. In both cases an internal recirculation zone was present. The laminar case is well predicted when account is taken of the estimated radial velocity component at the chosen inlet plane. The quadratic upwind interpolation and skew upwind schemes predict the main features of the internal recirculation zone also with a coarse grid. The turbulent case is well predicted with the coarse as well as the finer grids, the skew upwind and quadratic upwind interpolation schemes yielding results very close to the measurements. It is concluded that the skew upwind scheme reaches grid independence slightly before the quadratic upwind scheme, both considerably earlier than the hybrid scheme.     

钝锥三维粘性绕流背风面分离的数值模拟

本文将作者在文献[1]中提出的方法推广应用于求解三维可压缩 N-S 方程和简化 N-S 方程,并对近似因式分解法应用于三维问题的稳定性进行了分析。指出,对二维问题原无条件稳定的格式,经近似因式分解后仍是无条件稳定的;对于三维问题,原无条件稳定的格式经普通近似因式分解后所得到的格式可能是不稳定的或条件稳定的。利用系数矩阵分裂法所得到的近似因式分解格式可仍是无条件稳定的,只要适当加大分裂后的系数反差即可。 文中给出了钝锥超音速三维粘性绕流结果。得到了背风面分离的流动图像,物面压力值与实验值吻合。     

Effect of spatial discretization schemes on numerical solutions of viscoelastic fluid flows

This study examines the effect of discretization schemes for the convection term in the constitutive equation on numerical solutions of viscoelastic fluid flows. For this purpose, a temporally evolving mixing layer, a two-dimensional vortex pair interacting with a wall, and a fully developed turbulent channel flow are selected as test cases, and eight different discretization schemes are considered. Among them, the first-order upwind difference scheme (UD) and artificial diffusion scheme (AD), which are commonly used in the literature, show most stable and smooth solutions even for highly extensional flows. However, the stress fields are smeared too much by these schemes and the corresponding flow fields are quite different from those obtained by higher-order upwind difference schemes. Among higher-order upwind difference schemes investigated in this study, a third-order compact upwind difference scheme (CUD3) with locally added AD shows stable and most accurate solutions for highly extensional flows even at relatively high Weissenberg numbers.     

Third- and fourth-order well-balanced schemes for the shallow water equations based on the CWENO reconstruction

High-order finite volume schemes for conservation laws are very useful in applications, due to their ability to compute accurate solutions on quite coarse meshes and with very few restrictions on the kind of cells employed in the discretization. For balance laws, the ability to approximate up to machine precision relevant steady states allows the scheme to compute accurately, also on coarse meshes, small perturbations of such states, which are very relevant for many applications. In this paper, we propose third- and fourth-order accurate finite volume schemes for the shallow water equations. The schemes have the well-balanced property thanks to a path-conservative approach applied to an appropriate nonconservative reformulation of the equations. High-order accuracy is achieved by designing truly two-dimensional (2D) reconstruction procedures of the central WENO (CWENO ) type. The novel schemes are tested for accuracy and well-balancing and shown to maintain positivity of the water height on wet/dry transitions. Finally, they are applied to simulate the Tohoku 2011 tsunami event.     

Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.