A volume-agglomeration multirate time advancing for high Reynolds number flow simulation

A frequent configuration in computational fluid mechanics combines an explicit time advancing scheme for accuracy purposes and a computational grid with a very small portion of much smaller elements than in the remaining mesh. Two examples of such situations are the travel of a discontinuity followed by a moving mesh, and the large eddy simulation of high Reynolds number flows around bluff bodies where together very thin boundary layers and vortices of much more important size need to be captured. For such configurations, multistage explicit time advancing schemes with global time stepping are very accurate but very CPU consuming. In order to reduce this problem, the multirate (MR) time stepping approach represents an interesting improvement. The objective of such schemes, which allow to use different time steps in the computational domain, is to avoid penalizing the computational cost of the time advancement of unsteady solutions that would become large due to the use of small global time steps imposed by the smallest elements such as those constituting the boundary layers. In the present work, a new MR scheme based on control volume agglomeration is proposed for the solution of the compressible Navier-Stokes equations equipped with turbulence models. The method relies on a prediction step where large time steps are performed with an evaluation of the fluxes on macrocells for the smaller elements for stability purpose and a correction step in which small time steps are employed. The accuracy and efficiency of the proposed method are evaluated on several benchmarks flows: the problem of a moving contact discontinuity (inviscid flow), the computation with a hybrid turbulence model of flows around bluff bodies like a flow around a space probe model at Reynolds number 10⁶, a circular cylinder at Reynolds number 8.4 × 10⁶, and two tandem cylinders at Reynolds number 1.66 × 10⁵ and 1.4 × 10⁵.     

The double-tree method: An O(n) unsteady aerodynamic lifting surface method

Two new methods for reducing the computational cost of the unsteady vortex lattice method are developed. These methods use agglomeration to construct time-saving tree structures by approximating the effect of either a group of vortex rings or query points. A case study shows that combining the two new O(n·log n) tree methods together results in an O(n) method, called the double-tree method. Other case studies show that the trade-off between accuracy and speed can be easily and reliably controlled by the agglomeration cutoff distance. For a flat plate with 5 × 200 panels analyzed over 20 time steps, the double-tree method is 7 times faster than the unsteady vortex lattice method with a <5% difference in the force distribution and total lift coefficient. The case studies suggest that the computational benefit will increase for the same level of accuracy if the size of the problem is increased, making the method beneficial for full-aircraft analysis within optimization or dynamic load analysis, where the computational cost of the unsteady vortex lattice method can be large.     

On numerically predicting the onset and mode of instability in atomistic systems

We present an accurate and efficient method based on the Lanczos algorithm for predicting the onset and mode of instability in atomistic systems. Specifically, we develop a framework that is identically applicable to all flavors of atomistic simulations, including ab-initio calculations. Notably, we do not make any apriori assumptions regarding the nature of the instability or its location. We verify the accuracy of the proposed approach by studying defect nucleation during the nanoindentation of a triangular lattice and hydrostatic tension test of an aluminum crystal. We demonstrate that the computational cost in practical calculations scales linearly with system size, and is accompanied by a small prefactor. Overall, the proposed method is attractive because it enables the stability analysis of atomistic systems at the mesoscale.     

改进的物理融合神经网络在瑞利?泰勒不稳定性问题中的应用

基于相场法的物理融合神经网络PF-PINNs被成功用于两相流动的建模, 为两相流动的高精度直接数值模拟提供了全新的技术手段. 相场法作为一种新兴的界面捕捉方法, 其引入确保了界面的质量守恒, 显著提高了相界面的捕捉精度; 但是相场法中高阶导数的存在也降低了神经网络的训练速度. 为了提升计算训练过程的效率, 本文在PF-PINNS框架下, 参考深度混合残差方法MIM, 将化学能作为辅助变量以及神经网络的输出之一, 并修改了物理约束项的形式, 使辅助变量与相分数的关系式由硬约束转为了软约束. 上述两点改进显著降低了自动微分过程中计算图的规模, 节约了求导过程中的计算开销. 同时, 为了评估建立的PF-PINNS在雷诺数较高、计算量较大的场景中的建模能力, 本文将瑞利?泰勒RT不稳定性问题作为验证算例. 与高精度谱元法的定性与定量对比结果表明, 改进PF-PINNs有能力捕捉到两相界面的强非线性演化过程, 且计算精度接近传统算法, 计算结果符合物理规律. 改进前后的对比结果表明, 深度混合残差方法能够显著降低PF-PINNS的训练用时. 本文所述方法是进一步提升神经网络训练速度的重要参考资料, 并为探索高精度智能建模方法提供了全新的见解.      

一类高效率高分辨率加映射的WENO格式及其在复杂流动问题数值模拟中的应用

由于映射操作会带来额外的计算时间消耗, 传统加映射的WENO格式存在计算效率低的缺陷. 为了提高传统加映射WENO格式的计算效率, 通过利用标准符号函数的一种近似逼近函数构造出一族近似常值映射函数, 本文提出了一种新的加映射WENO格式, 称为WENO-ACM. 新映射函数满足文献中已有WENO-PM6格式映射函数的全部设计要求, 其中WENO-PM6是一种为了克服经典WENO-M格式潜在的精度丢失缺陷而提出的格式. 新格式保留了WENO-PM6在低耗散和高分辨率方面的优势, 同时, 显著的减少了每个时间步映射过程中的数学运算操作数, 进而在计算效率方面获得了明显的提升. 理论分析表明, 新格式在即使包含临界点的光滑区域也能够获得最佳收敛精度. 对近似色散关系的研究表明, 新格式的频谱特性也得到了显著的提升. 对大量标准测试算例进行了模拟计算, 包括精度测试、激波管问题、激波?熵波相互作用、爆炸波相互碰撞、二维黎曼问题、双马赫反射、前台阶流动、瑞利-泰勒不稳定性和开尔文?亥姆霍兹不稳定性问题等. 与广泛认可的WENO-JS, WENO-M, WENO-PM6格式综合比较发现, 新提出的WENO-ACM格式在高效率、低数值耗散和间断捕捉等方面都有显著的改进. 最重要的是, 与WENO-M和WENO-PM6格式相比, WENO-ACM将相对于WENO-JS格式的额外计算时间消耗分别减少了80%和90%以上.      

基于WENO-THINC/WLIC模型的水气二相流数值模拟

水气二相流与诸多领域的实际工程问题密切相关. 对二相流运动进行高精度的数值模拟是计算流体力学研究的难点和热点. 针对开敞水域的自由表面流运动问题, 将水和空气均视为不可压缩流体, 采用五阶加权基本无震荡(weighted essentially non-oscillatory, WENO)格式求解描述流体运动的纳维斯托克斯(Navier-Stokes, NS)方程, 利用以加权线性界面算法改进的多维双曲正切函数界面捕捉法(tangent of hyperbola for interface capturing with weighed line interface calculation, THINC/WLIC)追踪水气界面, 建立WENO-THINC/WLIC水气二相流运动数值模型. 模型采用分步计算法离散求解控制方程, 通过压力投影法求解压强场, 并利用三阶总变差递减(total variation diminishing, TVD)龙格库塔(Runge-Kutta, RK)法对时间项进行离散求解. 通过对环境速度场下Zalesak's disk和shearing vortex界面运动问题, 线性液舱晃荡问题以及溃坝问题的模拟结果与理论分析或试验结果的比较, 对所建立的水气二相流数值模型的适用性及模拟精度进行了验证. 结果表明, 本模型的模拟结果与物理模型或理论分析结果吻合良好, 能较为准确地再现不可压缩水气二相流运动现象. 鉴于WENO格式和THINC法本身在算法及应用等方面仍在不断改进, 本研究提出的WENO-THINC耦合模型为后续更高精度的二相流计算模型开发提供了一种研究思路.      

A Runge-Kutta–based WENO reconstruction on moving mesh for compressible fluids

In this paper, we construct a high-order moving mesh method based on total variation diminishing Runge-Kutta and weighted essential nonoscillatory reconstruction for compressible fluid system. Beginning with the integral form of fluid system, we get the semidiscrete system with an arbitrary mesh velocity. We use weighted essential nonoscillatory reconstruction to get the space accuracy on moving meshes, and the time accuracy is obtained by modified Runge-Kutta method; the mesh velocity is determined by moving mesh method. One- and two-dimensional numerical examples are presented to demonstrate the efficient and accurate performance of the scheme.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

Improved MPS method with variable‐size particles

Using variable‐size particles in the moving particle semi‐implicit method (MPS) could lead to inaccurate predictions and/or numerical instability. In this paper, a variable‐size particle moving particle semi‐implicit method (VSP‐MPS) scheme is proposed for the MPS method to achieve more reliable simulations with variable‐size particles. To improve stability and accuracy, a new gradient model is developed based on a previously developed MPS scheme that requires no surface detection MPS. The dynamic particle coalescing and splitting algorithm is revised to achieve dynamic multi‐resolution. A cubic spline function with additional function is employed as the kernel function. The effectiveness of the VSP‐MPS method is demonstrated by three verification examples, that is, a hydrostatic pressure problem, a complicated free surface flow problem with large deformation, and a dynamic impact problem. The new VSP‐MPS scheme with variable‐size particles is found to have balanced efficiency and accuracy that is suitable for simulating large systems with complex flow patterns. Copyright © 2015 John Wiley & Sons, Ltd.     

高Bond数下黏性液滴的Rayleigh-Taylor不稳定性

给出了高Bond数下黏性液滴表面Rayleigh-Taylor线性不稳定性的分析解，这种不稳定性对于超音速气流作用下液滴破碎的早期阶段起着至关重要的作用．基于稳定性分析的结果，导出了用于估算稳定液滴的最大直径及液滴无量纲初始破碎时间的计算式，这些计算式与相关文献给出的实验和分析结果比较显示了良好的一致．     

An ALE method for simulations of axisymmetric elastic surfaces in flow

The dynamics of membranes, shells, and capsules in fluid flow has become an active research area in computational physics and computational biology. The small thickness of these elastic materials enables their efficient approximation as a hypersurface, which exhibits an elastic response to in-plane stretching and out-of-plane bending, possibly accompanied by a surface tension force. In this work, we present a novel arbitrary Lagrangian-Eulerian (ALE) method to simulate such elastic surfaces immersed in Navier-Stokes fluids. The method combines high accuracy with computational efficiency, since the grid is matched to the elastic surface and can, therefore, be resolved with relatively few grid points. The focus of this work is on axisymmetric shapes and flow conditions, which are present in a wide range of biophysical problems. We formulate axisymmetric elastic surface forces and propose a discretization with surface finite-differences coupled to evolving finite elements. We further develop an implicit coupling strategy to reduce time step restrictions. We show in several numerical test cases that accurate results can be achieved at computational times on the order of minutes on a single core CPU. We demonstrate two state-of-the-art applications which to our knowledge cannot be simulated with any other numerical method so far: we present first simulations of the observed shape oscillations of novel microswimming shells and the uniaxial compression of the cortex of a biological cell during an AFM experiment.     

Improved CVP scheme for laminar incompressible flows

An improved scheme of the continuity vorticity pressure (CVP) variational equations method is presented. The changes from the original version of the CVP method concern the velocity and the pressure correction equations that are used in the solution procedure and the topology of the grid where the method is applied. The improved CVP scheme is faster, simpler and more stable than the original version of the method. The efficiency and the accuracy of the new scheme are tested and validated through comparison of predictions and of computational time, with numerical results obtained with the SIMPLE method. Moreover, we present extensive comparisons of the results of the improved CVP scheme with numerical and experimental data from various researchers that show excellent agreement for a wide range of benchmark 2D and 3D laminar internal flow problems such as flow over a backward facing step, flow in square, circular and elliptical curved ducts and pulsating flow. Copyright © 2010 John Wiley & Sons, Ltd.     

An attempt to improve accuracy of higher‐order statistics and spectra in direct numerical simulation of incompressible wall turbulence by using the compact schemes for viscous terms

We attempt to improve accuracy in the high‐wavenumber region in DNS of incompressible wall turbulence such as found in fully developed turbulent channel flow. In particular, it is shown that the improvement of accuracy of viscous terms in the Navier–Stokes equations leads to the improvement of accuracy of higher‐order statistics and various spectra. It is emphasized that increase in required computational cost will not be crucial when incompressible flow is simulated, because the introduction of a higher‐order scheme into the viscous terms does not increase computational cost for solving the Poisson equation. We introduced fourth‐order and eighth‐order central compact schemes for discretizing the viscous terms in DNS of a fully developed turbulent channel flow. The results are compared with those using second‐order and fourth‐order central‐difference schemes applied to the viscous terms and those obtained by the spectral method. The results show that accuracy improvement of the viscous terms improve accuracy of higher‐order statistics (i.e., skewness and flatness factors of streamwise velocity fluctuation) and various spectra of velocity and pressure fluctuations in the high‐wavenumber region. Copyright © 2013 John Wiley & Sons, Ltd.     

Linear Rayleigh-Taylor instability analysis of double-shell Kidder＇s self-similar implosion solution

This paper generalizes the single-shell Kidder＇s self-similar solution to the double-shell one with a discontinuity in density across the interface. An isentropic implosion model is constructed to study the Rayleigh-Taylor instability for the implosion compression. A Godunov-type method in the Lagrangian coordinates is used to compute the one-dimensional Euler equation with the initial and boundary conditions for the double-shell Kidder＇s self-similar solution in spherical geometry. Numerical results are obtained to validate the double-shell implosion model. By programming and using the linear perturbation codes, a linear stability analysis on the Rayleigh-Taylor instability for the double-shell isentropic implosion model is performed. It is found that, when the initial perturbation is concentrated much closer to the interface of the two shells, or when the spherical wave number becomes much smaller, the modal radius of the interface grows much faster, i.e., more unstable. In addition, from the spatial point of view for the compressibility effect on the perturbation evolution, the compressibility of the outer shell has a destabilization effect on the Rayleigh-Taylor instability, while the compressibility of the inner shell has a stabilization effect.     

A second‐order time accurate finite element method for quasi‐incompressible viscous flows

A finite element method for quasi‐incompressible viscous flows is presented. An equation for pressure is derived from a second‐order time accurate Taylor–Galerkin procedure that combines the mass and the momentum conservation laws. At each time step, once the pressure has been determined, the velocity field is computed solving discretized equations obtained from another second‐order time accurate scheme and a least‐squares minimization of spatial momentum residuals. The terms that stabilize the finite element method (controlling wiggles and circumventing the Babuska–Brezzi condition) arise naturally from the process, rather than being introduced a priori in the variational formulation. A comparison between the present second‐order accurate method and our previous first‐order accurate formulation is shown. The method is also demonstrated in the computation of the leaky‐lid driven cavity flow and in the simulation of a crossflow past a circular cylinder. In both cases, good agreement with previously published experimental and computational results has been obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

A Galerkin finite element algorithm based on third-order Runge-Kutta temporal discretization along the uniform streamline for unsteady incompressible flows

In this paper, for two-dimensional unsteady incompressible flow, the Navier-Stokes equations without convection term are derived by the coordinate transformation along the streamline characteristic. The third-order Runge-Kutta method along the streamline is introduced to discrete the alternative Navier-Stokes equations in time, and spacial discretization is carried out by the Galerkin method, and then, the third-order accuracy finite element method is obtained. Meanwhile, the streamline velocity is uniformly approximated by initial velocity in each time step in order to reduce update frequency of total element matrix and improve calculation efficiency. Finally, some classic unsteady flow examples are calculated and analyzed by different calculation methods, which further demonstrate that the present method has more advantages in stability, permissible time step, dissipation, computational cost, and accuracy. The code can be downloaded at https://doi.org/10.13140/RG.2.2.27706.44484 .     

Analysis of efficient preconditioned defect correction methods for nonlinear water waves

Robust computational procedures for the solution of non‐hydrostatic, free surface, irrotational and inviscid free‐surface water waves in three space dimensions can be based on iterative preconditioned defect correction (PDC) methods. Such methods can be made efficient and scalable to enable prediction of free‐surface wave transformation and accurate wave kinematics in both deep and shallow waters in large marine areas or for predicting the outcome of experiments in large numerical wave tanks. We revisit the classical governing equations are fully nonlinear and dispersive potential flow equations. We present new detailed fundamental analysis using finite‐amplitude wave solutions for iterative solvers. We demonstrate that the PDC method in combination with a high‐order discretization method enables efficient and scalable solution of the linear system of equations arising in potential flow models. Our study is particularly relevant for fast and efficient simulation of non‐breaking fully nonlinear water waves over varying bottom topography that may be limited by computational resources or requirements. To gain insight into algorithmic properties and proper choices of discretization parameters for different PDC strategies, we study systematically limits of accuracy, convergence rate, algorithmic and numerical efficiency and scalability of the most efficient known PDC methods. These strategies are of interest, because they enable generalization of geometric multigrid methods to high‐order accurate discretizations and enable significant improvement in numerical efficiency while incuring minimal storage requirements. We demonstrate robustness using such PDC methods for practical ranges of interest for coastal and maritime engineering, that is, from shallow to deep water, and report details of numerical experiments that can be used for benchmarking purposes. Copyright © 2014 John Wiley & Sons, Ltd.     

modeling of multicomponent multiphase mixture flows on the basis of the density-functional method

Examples of numerical calculations of isothermal flows of two-phase two-component mixtures based on the density-functional method are presented. Using this method, the following problems are calculated in the two-dimensional formulation: drop impact on a liquid layer, drop rupture in a Couette flowfield, wetting-angle formation for a drop on a solid surface, development of Rayleigh-Taylor and Kelvin-Helmholtz instability on a gas-liquid interface.Translated from Izvestiya Rossiiskoi Academii Nauk, Mekhanika Zhidkosti i Gaza, No. 6, 2004, pp. 101–114.Original Russian Text Copyright © 2004 by Demyanov and Dinariev.     

Operator-splitting methods for the incompressible Navier-Stokes equations on non-staggered grids. Part 1: First-order schemes

New implicit finite difference schemes for solving the time-dependent incompressible Navier-Stokes equations using primitive variables and non-staggered grids are presented in this paper. A priori estimates for the discrete solution of the methods are obtained. Employing the operator approach, some requirements on the difference operators of the scheme are formulated in order to derive a scheme which is essentially consistent with the initial differential equations. The operators of the scheme inherit the fundamental properties of the corresponding differential operators and this allows a priori estimates for the discrete solution to be obtained. The estimate is similar to the corresponding one for the solution of the differential problem and guarantees boundedness of the solution. To derive the consistent scheme, special approximations for convective terms and div and grad operators are employed. Two variants of time discretization by the operator-splitting technique are considered and compared. It is shown that the derived scheme has a very weak restriction on the time step size. A lid-driven cavity flow has been predicted to examine the stability and accuracy of the schemes for Reynolds number up to 3200 on the sequence of grids with 21 × 21, 41 × 41, 81 × 81 and 161 × 161 grid points.     

Effect of initial disturbances on the Rayleigh-Taylor instability in a Hele-Shaw cell

The effect of the initial state on the development of the Rayleigh-Taylor instability in a Hele-Shaw cell is studied. The initial disturbances were initiated using two techniques, namely, by exciting the standing Faraday waves by means of a vibrator and by periodically distorting the interface between the fluids by means of thin connecting needles. The Rayleigh-Taylor instability was realized by inverting the cell. The process of wave development at the interface between the fluids was filmed with subsequent computer processing of the videofilm obtained. It was found that the initial state is the decisive factor in the development of the process up to large times.