A robust and well‐balanced scheme for the 2D Saint‐Venant system on unstructured meshes with friction source term

In the following lines, we propose a numerical scheme for the shallow‐water system supplemented by topography and friction source terms, in a 2D unstructured context. This work proposes an improved version of the well‐balanced and robust numerical model recently introduced by Duran et al. (J. Comp. Phys., 235 , 565–586, 2013) for the pre‐balanced shallow‐water equations, accounting for varying topography. The present work aims at relaxing the robustness condition and includes a friction term. To this purpose, the scheme is modified using a recent method, entirely based on a modified Riemann solver. This approach preserves the robustness and well‐balanced properties of the original scheme and prevents unstable computations in the presence of low water depths. A series of numerical experiments are devoted to highlighting the performances of the resulting scheme. Simulations involving dry areas, complex geometry and topography are proposed to validate the stability of the numerical model in the neighbourhood of wet/dry transitions. Copyright © 2015 John Wiley & Sons, Ltd.     

MFCAV近似Riemann解在新型拉氏方法中的应用

Maire等提出了一种新型的有限体积中心型拉氏方法, 该方法大大地改善了一直困扰着一般中心型拉氏方法的虚假网格变形. 然而在计算数值流和移动网格时,该方法只应用了数值黏性较大的弱波近似(weak wave approximatedmethod, WWAM) Riemann解, 而且方法的设计表明其他类型的近似Riemann解不能简单直接地应用上去. 将体平均多流管(multifluid channel on averaged volume, MFCAV)近似Riemann解视为对WWAM的修正,成功将其应用于新型方法中, 数值实验表明应用了MFCAV 的新方法是有效的. 研究为将其他更为有效的近似Riemann解应用于该新型方法中开辟了一条道路.      

Monte Carlo method of radiative transfer applied to a turbulent flame modeling with LES

Radiative transfer plays an important role in the numerical simulation of turbulent combustion. However, for the reason that combustion and radiation are characterized by different time scales and different spatial and chemical treatments, the radiation effect is often neglected or roughly modelled. The coupling of a large eddy simulation combustion solver and a radiation solver through a dedicated language, CORBA, is investigated. Two formulations of Monte Carlo method (Forward Method and Emission Reciprocity Method) employed to resolve RTE have been compared in a one-dimensional flame test case using three-dimensional calculation grids with absorbing and emitting media in order to validate the Monte Carlo radiative solver and to choose the most efficient model for coupling. Then the results obtained using two different RTE solvers (Reciprocity Monte Carlo method and Discrete Ordinate Method) applied on a three-dimensional flame holder set-up with a correlated-k distribution model describing the real gas medium spectral radiative properties are compared not only in terms of the physical behavior of the flame, but also in computational performance (storage requirement, CPU time and parallelization efficiency). To cite this article: J. Zhang et al., C. R. Mecanique 337 (2009).     

Improved atomization,collision and sub-grid scale momentum coupling models for transient vaporizing engine sprays

A computationally efficient spray model is presented for the simulation of transient vaporizing engine sprays. It is applied to simulate high-pressure fuel injections in a constant volume chamber and in mixture preparation experiments in a light-duty internal combustion engine. The model is based on the Lagrangian-Particle/Eulerian-Fluid approach, and an improved blob injection model is used that removes numerical dependency on the injected number of computational parcels. Atomization is modeled with the hybrid Kelvin–Helmholtz/Rayleigh–Taylor scheme, in combination with a drop drag model that includes Mach number and Knudsen number effects. A computationally efficient drop collision scheme is presented, tailored for large numbers of parcels, using a deterministic collision impact definition and kd-tree data search structure to perform radius-of-influence based, grid-independent collision probability estimations. A near-nozzle sub-grid scale flow-field representation is introduced to reduce numerical grid dependency, which uses a turbulent transient gas-jet model with a Stokes–Strouhal analogy assumption. An implicit coupling method was developed for the Arbitrary Lagrangian–Eulerian (ALE) turbulent flow solver. A multi-objective genetic algorithm was used to study the interactions of the various model constants, and to provide an optimal calibration. The optimal set showed similar values of the primary breakup constants as values used in the literature. However, different values were seen for the gas-jet model constants for accurate simulations of the initial spray transient. The results show that there is a direct correlation between the predicted initial liquid-phase transient and the global gas-phase jet penetration. Model validation was also performed in engine simulations with the same set of constants. The model captured mixture preparation well in all cases, proving its suitability for simulations of transient spray injection in engines.     

Dilation‐based shock capturing for high‐order methods

In this paper, we introduce a shock‐capturing artificial viscosity technique for high‐order unstructured mesh methods. This artificial viscosity model is based on a non‐dimensional form of the divergence of the velocity. The technique is an extension and improvement of the dilation‐based artificial viscosity methods introduced in Premasuthan et al., 15 and further extended in Nguyen and Peraire 27 . The approach presented has a number attractive properties including non‐dimensional analytical form, sub‐cell resolution, and robustness for complex shock flows on anisotropic meshes. We present extensive numerical results to demonstrate the performance of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.     

Algebraic multigrid and incompressible fluid flow

This paper is concerned with the development of algebraic multigrid (AMG) solution methods for the coupled vector–scalar fields of incompressible fluid flow. It addresses in particular the problems of unstable smoothing and of maintaining good vector–scalar coupling in the AMG coarse‐grid approximations. Two different approaches have been adopted. The first is a direct approach based on a second‐order discrete‐difference formulation in primitive variables. Here smoothing is stabilized using a minimum residual control harness and velocity–pressure coupling is maintained by employing a special interpolation during the construction of the inter‐grid transfer operators. The second is an indirect approach that avoids the coupling problem altogether by using a fourth‐order discrete‐difference formulation in a single scalar‐field variable, primitive variables being recovered in post‐processing steps. In both approaches the discrete‐difference equations are for the steady‐state limit (infinite time step) with a fully implicit treatment of advection based on central differencing using uniform and non‐uniform unstructured meshes. They are solved by Picard iteration, the AMG solvers being used repeatedly for each linear approximation. Both classical AMG (C‐AMG) and smoothed‐aggregation AMG (SA‐AMG) are used. In the direct approach, the SA‐AMG solver (with inter‐grid transfer operators based on mixed‐order interpolation) provides an almost mesh‐independent convergence. In the indirect approach for uniform meshes, the C‐AMG solver (based on a Jacobi‐relaxed interpolation) provides solutions with an optimum scaling of the convergence rates. For non‐uniform meshes this convergence becomes mesh dependent but the overall solution cost increases relatively slowly with increasing mesh bandwidth. Copyright © 2006 John Wiley & Sons, Ltd.     

An Anisotropic A posteriori Error Estimator for CFD

In this article, a robust anisotropic adaptive algorithm is presented, to solve compressible-flow equations using a stabilized CFD solver and automatic mesh generators. The association includes a mesh generator, a flow solver, and an a posteriori error-estimator code. The estimator was selected among several choices available (Almeida et al. (2000). Comput. Methods Appl. Mech. Engng , 182 , 379-400; Borges et al. (1998). "Computational mechanics: new trends and applications". Proceedings of the 4th World Congress on Computational Mechanics , Bs.As., Argentina) giving a powerful computational tool. The main aim is to capture solution discontinuities, in this case, shocks, using the least amount of computational resources, i.e. elements, compatible with a solution of good quality. This leads to high aspect-ratio elements (stretching). To achieve this, a directional error estimator was specifically selected. The numerical results show good behavior of the error estimator, resulting in strongly-adapted meshes in few steps, typically three or four iterations, enough to capture shocks using a moderate and well-distributed amount of elements.     

Multigrid flow solutions in complex two-dimensional geometries

A finite difference solution algorithm is described for use on two-dimensional curvilinear meshes generated by the solution of the transformed Laplace equation. The efficiency of the algorithm is improved through the use of a full approximation scheme (FAS) multigrid algorithm using an extended pressure correction scheme as smoother. The multigrid algorithm is implemented as a fixed V-cycle through the grid levels with a constant number of sweeps being performed at each grid level. The accuracy and efficiency of the numerical code are validated using comparisons of the flow over two backward step configurations. Results show close agreement with previous numerical predictions and experimental data. Using a standard Cartesian co-ordinate flow solver, the multigrid efficiency obtainable in a rectangular system is shown to be reproducible in two-dimensional body-fitted curvilinear co-ordinates. Comparisons with a standard one-grid method show the multigrid method, on curvilinear meshes, to give reductions in CPU time of up to 93%.     

Direct simulation of flows with suspended paramagnetic particles using one-stage smoothed profile method

The so-called smoothed profile method, originally suggested by Nakayama and Yamamoto and further improved by Luo et al. in 2005 and 2009, respectively, is an efficient numerical solver for fluid-structure interaction problems, which represents the particles by a certain smoothed profile on a fixed grid and constructs some form of body force added into the momentum (Navier-Stokes) equation by ensuring the rigidity of particles. For numerical simulations, the method first advances the flow and pressure fields by integrating the momentum equation except the body-force (momentum impulse) term in time and next updates them by separately taking temporal integration of the body-force term, thus requiring one more Poisson-equation solver for the extra pressure field due to the rigidity of particles to ensure the divergence-free constraint of the total velocity field. In the present study, we propose a simplified version of the smoothed profile method or the one-stage method, which combines the two stages of velocity update (temporal integration) into one to eliminate the necessity for the additional solver and, thus, significantly save the computational cost. To validate the proposed one-stage method, we perform the so-called direct numerical simulations on the two-dimensional motion of multiple inertialess paramagnetic particles in a nonmagnetic fluid subjected to an external uniform magnetic field and compare their results with the existing benchmark solutions. For the validation, we develop the finite-volume version of the direct simulation method by employing the proposed one-stage method. Comparison shows that the proposed one-stage method is very accurate and efficient in direct simulations of such magnetic particulate flows.     

A well‐balanced stable generalized Riemann problem scheme for shallow water equations using adaptive moving unstructured triangular meshes

We propose a well‐balanced stable generalized Riemann problem (GRP) scheme for the shallow water equations with irregular bottom topography based on moving, adaptive, unstructured, triangular meshes. In order to stabilize the computations near equilibria, we use the Rankine–Hugoniot condition to remove a singularity from the GRP solver. Moreover, we develop a remapping onto the new mesh (after grid movement) based on equilibrium variables. This, together with the already established techniques, guarantees the well‐balancing. Numerical tests show the accuracy, efficiency, and robustness of the GRP moving mesh method: lake at rest solutions are preserved even when the underlying mesh is moving (e.g., mesh points are moved to regions of steep gradients), and various comparisons with fixed coarse and fine meshes demonstrate high resolution at relatively low cost. Copyright © 2013 John Wiley & Sons, Ltd.     

基于计算流体力学的“虚拟飞行”技术及初步应用

以动态混合网格技术为基础, 通过耦合求解刚体动力学方程、流体力学控制方程以及飞行控制律, 建立了适用于飞行器"虚拟飞行" 过程研究的一体化数值模拟技术. 通过典型的外挂物投放算例对流体力学控制方程/动力学方程的耦合算法进行了测试, 并对某导弹的姿态角控制过程、过载控制过程以及变马赫数条件下的控制过程进行了数值模拟, 得到了与实验非常一致的结果. 这些算例证明该一体化算法已经初步具备了针对复杂飞行器"数值虚拟飞行" 的应用能力.      

A Cartesian cut cell based two-way strong fluid–solid coupling algorithm for 2D floating bodies

In a recent paper Kelly et al. (2015) [PICIN: A Particle-In-Cell solver for incompressible free surface flows with two-way fluid–solid coupling. SIAM Journal on Scientific Computing 37 (3), B403–24.] detailed the PICIN full particle Particle-In-Cell (PIC) solver for incompressible free-surface flows. The model described in that paper employed a tailored version of the Distributed Lagrange Multiplier (DLM) method for the strong coupling of fluid–solid interaction. In this paper we propose an alternative strong fluid–solid coupling algorithm based on a modification to the cut cell methodology that is informed by the variational approach. The solid velocity flux/integral on the boundary is expressed purely in terms of pressure leading to a revised pressure Poisson equation that is discretised in a finite volume sense. This approach allows the PICIN model to simulate the motion of floating bodies of arbitrary configuration. 2D test cases involving floating bodies with one or more degrees of freedom (DoF) are used to validate the modified PICIN model. The results presented show that the modified PICIN model is able to both efficiently and robustly predict the motions of surface-piercing floating structures under either regular or extreme wave action.     

Analysis of preconditioned iterative solvers for incompressible flow problems

Solving efficiently the incompressible Navier–Stokes equations is a major challenge, especially in the three‐dimensional case. The approach investigated by Elman et al. (Finite Elements and Fast Iterative Solvers. Oxford University Press: Oxford, 2005) consists in applying a preconditioned GMRES method to the linearized problem at each iteration of a nonlinear scheme. The preconditioner is built as an approximation of an ideal block‐preconditioner that guarantees convergence in 2 or 3 iterations. In this paper, we investigate the numerical behavior for the three‐dimensional lid‐driven cavity problem with wedge elements; the ultimate motivation of this analysis is indeed the development of a preconditioned Krylov solver for stratified oceanic flows which can be efficiently tackled using such meshes. Numerical results for steady‐state solutions of both the Stokes and the Navier–Stokes problems are presented. Theoretical bounds on the spectrum and the rate of convergence appear to be in agreement with the numerical experiments. Sensitivity analysis on different aspects of the structure of the preconditioner and the block decomposition strategies are also discussed. Copyright © 2011 John Wiley & Sons, Ltd.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.     

Application of the GSMAC-CIP Method to Incompressible Navier-Stokes Equations at High Reynolds Numbers

The conventional shape function for the finite-element method (FEM) is linear, and it is thus inadequate for analyzing numerically complex flows at high Reynolds numbers. In this study, we propose a new scheme, GSMAC-CIP, using the third-order shape function, which requires continuity of the value of the function and its first space derivative in the whole space and is formulated by a finite element method for the cubic interpolated pseudo-particle (CIP) method. We verified the effectiveness of this new scheme by analyzing the forced-driven convection in a square cavity at Re = 1000, 5000 and 10000. The numerical results obtained by the present scheme are compared with those of GSMAC-FEM using coarser meshes, and it is shown that the present scheme is superior to GSMAC-FEM in terms of space accuracy. Moreover, it is shown that the numerical results obtained by the present scheme using fine meshes were in precise agreement with those obtained by Ghia et al.     

一种简单的精确捕捉接触间断的黎曼求解器

基于Godunov型数值格式的有限体积法是求解双曲型守恒律系统的主流方法,其中用来计算界面数值通量的黎曼求解器在很大程度上决定了数值格式在计算中的表现。单波的Rusanov求解器和双波的HLL求解器具有简单、高效和鲁棒性好等优点,但是在捕捉接触间断时耗散太大。全波的HLLC格式能够精确捕捉接触间断,但是在计算中出现的激波不稳定现象限制了其在高马赫数流动问题中的应用。本文利用双曲正切函数和五阶WENO格式来重构界面两侧的密度值,并且结合边界变差下降算法来减小Rusanov格式耗散项中的密度差,从而提高格式对于接触间断的分辨率。研究表明,相比于全波的HLLC求解器,本文构造的黎曼求解器不仅具有更高的接触分辨率,而且还具有更好的激波稳定性。     

Large eddy simulation of incompressible free round jet with discontinuous Galerkin method

In the paper, discontinuous Galerkin method is applied to simulation of incompressible free round turbulent jet using large eddy simulation with eddy viscosity approach. The solution algorithm is based on the classical projection method, but instead of the solution of the Poisson equation, a parabolic equation is advanced in pseudo‐time, which provides the pressure field ensuring the proper pressure–velocity coupling. For time and pseudo‐time integration, explicit Runge–Kutta method is employed. The computational meshes consist of hexahedral elements with flat faces. Within a given finite element, all flow variables are expressed with modal expansions of the same order (including velocity and pressure). Discretisation of the viscous terms in the Navier–Stokes equations and Laplacian in the Poisson equation is stabilised with mixed finite element approach. The correctness of the solution algorithm is verified in a commonly used test case of laminar flow in 3D lid‐driven cavity. The results of computations of the free jet are compared with experimental and numerical reference data, the latter obtained from the high‐order pseudospectral code. The statistics of centerline flow velocity – mean velocity and its fluctuations – show satisfactory agreement with the reference data. Copyright © 2015 John Wiley & Sons, Ltd.     

High-order Lagrangian cell-centered conservative scheme on unstructured meshes

A high-order Lagrangian cell-centered conservative gas dynamics scheme is presented on unstructured meshes. A high-order piecewise pressure of the cell is intro- duced. With the high-order piecewise pressure of the cell, the high-order spatial discretiza- tion fluxes are constructed. The time discretization of the spatial fluxes is performed by means of the Taylor expansions of the spatial discretization fluxes. The vertex velocities are evaluated in a consistent manner due to an original solver located at the nodes by means of momentum conservation. Many numerical tests are presented to demonstrate the robustness and the accuracy of the scheme.