A high‐order flux reconstruction adaptive mesh refinement method for magnetohydrodynamics on unstructured grids

We report our recent development of the high‐order flux reconstruction adaptive mesh refinement (AMR) method for magnetohydrodynamics (MHD). The resulted framework features a shock‐capturing duo of AMR and artificial resistivity (AR), which can robustly capture shocks and rotational and contact discontinuities with a fraction of the cell counts that are usually required. In our previous paper, 36 we have presented a shock‐capturing framework on hydrodynamic problems with artificial diffusivity and AMR. Our AMR approach features a tree‐free, direct‐addressing approach in retrieving data across multiple levels of refinement. In this article, we report an extension to MHD systems that retains the flexibility of using unstructured grids. The challenges due to complex shock structures and divergence‐free constraint of magnetic field are more difficult to deal with than those of hydrodynamic systems. The accuracy of our solver hinges on 2 properties to achieve high‐order accuracy on MHD systems: removing the divergence error thoroughly and resolving discontinuities accurately. A hyperbolic divergence cleaning method with multiple subiterations is used for the first task. This method drives away the divergence error and preserves conservative forms of the governing equations. The subiteration can be accelerated by absorbing a pseudo time step into the wave speed coefficient, therefore enjoys a relaxed CFL condition. The AMR method rallies multiple levels of refined cells around various shock discontinuities, and it coordinates with the AR method to obtain sharp shock profiles. The physically consistent AR method localizes discontinuities and damps the spurious oscillation arising in the curl of the magnetic field. The effectiveness of the AMR and AR combination is demonstrated to be much more powerful than simply adding AR on finer and finer mesh, since the AMR steeply reduces the required amount of AR and confines the added artificial diffusivity and resistivity to a narrower and narrower region. We are able to verify the designed high‐order accuracy in space by using smooth flow test problems on unstructured grids. The efficiency and robustness of this framework are fully demonstrated through a number of two‐dimensional nonsmooth ideal MHD tests. We also successfully demonstrate that the AMR method can help significantly save computational cost for the Orszag‐Tang vortex problem.     

Accurate and robust adaptive mesh refinement for aerodynamic simulation with multi‐block structured curvilinear mesh

A multi‐block curvilinear mesh‐based adaptive mesh refinement (AMR) method is developed to satisfy the competing objectives of improving accuracy and reducing cost. Body‐fitted curvilinear mesh‐based AMR is used to capture flow details of various length scales. A series of efforts are made to guarantee the accuracy and robustness of the AMR system. A physics‐based refinement function is proposed, which is proved to be able to detect both shock wave and vortical flow. The curvilinear mesh is refined with cubic interpolation, which guarantees the aspect ratio and smoothness. Furthermore, to enable its application in complex configurations, a sub‐block‐based refinement strategy is developed to avoid generating invalid mesh, which is the consequence of non‐smooth mesh lines or singular geometry features. A newfound problem of smaller wall distance, which negatively affects the stability and is never reported in the literature, is also discussed in detail, and an improved strategy is proposed. Together with the high‐accuracy numerical scheme, a multi‐block curvilinear mesh‐based AMR system is developed. With a series of test cases, the current method is verified to be accurate and robust and be able to automatically capture the flow details at great cost saving compared with the global refinement. Copyright © 2015 John Wiley & Sons, Ltd.     

An approach of dynamic mesh adaptation for simulating 3‐dimensional unsteady moving‐immersed‐boundary flows

In this paper, we present an approach of dynamic mesh adaptation for simulating complex 3‐dimensional incompressible moving‐boundary flows by immersed boundary methods. Tetrahedral meshes are adapted by a hierarchical refining/coarsening algorithm. Regular refinement is accomplished by dividing 1 tetrahedron into 8 subcells, and irregular refinement is only for eliminating the hanging points. Merging the 8 subcells obtained by regular refinement, the mesh is coarsened. With hierarchical refining/coarsening, mesh adaptivity can be achieved by adjusting the mesh only 1 time for each adaptation period. The level difference between 2 neighboring cells never exceeds 1, and the geometrical quality of mesh does not degrade as the level of adaptive mesh increases. A predictor‐corrector scheme is introduced to eliminate the phase lag between adapted mesh and unsteady solution. The error caused by each solution transferring from the old mesh to the new adapted one is small because most of the nodes on the 2 meshes are coincident. An immersed boundary method named local domain‐free discretization is employed to solve the flow equations. Several numerical experiments have been conducted for 3‐dimensional incompressible moving‐boundary flows. By using the present approach, the number of mesh nodes is reduced greatly while the accuracy of solution can be preserved.     

An adaptive cut‐cell method for environmental fluid mechanics

In this work we present a numerical method for solving the incompressible Navier–Stokes equations in an environmental fluid mechanics context. The method is designed for the study of environmental flows that are multiscale, incompressible, variable‐density, and within arbitrarily complex and possibly anisotropic domains. The method is new because in this context we couple the embedded‐boundary (or cut‐cell) method for complex geometry with block‐structured adaptive mesh refinement (AMR) while maintaining conservation and second‐order accuracy. The accurate simulation of variable‐density fluids necessitates special care in formulating projection methods. This variable‐density formulation is well known for incompressible flows in unit‐aspect ratio domains, without AMR, and without complex geometry, but here we carefully present a new method that addresses the intersection of these issues. The methodology is based on a second‐order‐accurate projection method with high‐order‐accurate Godunov finite‐differencing, including slope limiting and a stable differencing of the nonlinear convection terms. The finite‐volume AMR discretizations are based on two‐way flux matching at refinement boundaries to obtain a conservative method that is second‐order accurate in solution error. The control volumes are formed by the intersection of the irregular embedded boundary with Cartesian grid cells. Unlike typical discretization methods, these control volumes naturally fit within parallelizable, disjoint‐block data structures, and permit dynamic AMR coarsening and refinement as the simulation progresses. We present two‐ and three‐dimensional numerical examples to illustrate the accuracy of the method. Copyright © 2008 John Wiley & Sons, Ltd.     

Adaptive non‐conformal mesh refinement and extended finite element method for viscous flow inside complex moving geometries

Two different techniques to analyze non‐Newtonian viscous flow in complex geometries with internal moving parts and narrow gaps are compared. The first technique is a non‐conforming mesh refinement approach based on the fictitious domain method (FDM), and the second one is the extended finite element method (XFEM). The refinement technique uses one fixed reference mesh, and to impose continuity across non‐conforming regions, constraints using Lagrangian multipliers are used. The size of elements locally in the high shear rate regions is reduced to increase accuracy. FDM is shown to have limitations; therefore, XFEM is applied to decouple the fluid from the internal moving rigid bodies. In XFEM, the discontinuous field variables are captured by using virtual degrees of freedom that serve as enrichment and by applying special integration over the intersected elements. The accuracy of the two methods is demonstrated by direct comparison with results of a boundary‐fitted mesh applied to a two‐dimensional cross section of a twin‐screw extruder. Compared with non‐conforming FDM, XFEM shows a considerable improvement in accuracy around the rigid body, especially in the narrow gap regions. Copyright © 2011 John Wiley & Sons, Ltd.     

Adaptive mesh refinement method-based large eddy simulation for the flow over circular cylinder at ReD = 3900

With the development of computational power, large eddy simulation (LES) method is increasingly used in simulating complex flow. However, there still exist many factors affecting the LES quality and appropriate mesh resolution is among one of them. This work aims to develop an automatic procedure to refine the LES mesh by combining adaptive mesh refinement (AMR) and LES quality criteria. An LES refinement criterion is developed by estimating the proper grid length scale which meets the accuracy requirement of LES method. With this criterion, the baseline mesh is automatically refined with the AMR method. In this work, an efficient one-shot refinement strategy is also proposed to reduce the overall simulation time. Current AMR-based LES method is verified with the typical LES test case about the flow past circular cylinder at Re _D = 3900. Results show that the automatically refined mesh provides systematically better agreement with experimental results and with current method the balance between accuracy and computational expense for LES can be obtained.     

Three‐dimensional transient free‐surface flow of viscous fluids inside cavities of arbitrary shape

The three‐dimensional transient free‐surface flow inside cavities of arbitrary shape is examined in this study. An adaptive (Lagrangian) boundary‐element approach is proposed for the general three‐dimensional simulation of confined free‐surface flow of viscous incompressible fluids. The method is stable as it includes remeshing capabilities of the deforming free‐surface, and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. The method is used to determine the flow field and free‐surface evolution inside cubic, rectangular and cylindrical containers. These problems illustrate the transient nature of the flow during the mixing process. Surface tension effects are also explored. Copyright © 2003 John Wiley & Sons, Ltd.     

A high‐resolution method for compressible two‐fluid flows and simulation of three‐dimensional shock–bubble interactions

A high‐resolution method is developed to capture the material interfaces of compressible two‐fluid flows in multiple dimensions. A fluid mixture model system with single velocity and pressure is used, and viscous effect can also be taken into account. A consistent thermodynamic law based on the assumption of pressure equilibrium is employed to describe the thermodynamic behaviors of the pure fluids and mixture of two components. The splitting and unsplit Eulerian formulations of piecewise parabolic method are extended to numerically integrate the hyperbolic part of the model system, whereas the system of diffusion equations is solved using an explicit, central difference scheme. The block‐structured adaptive mesh refinement (AMR) capability is built in the hydrodynamic code to locally improve grid resolution. The resulting method is verified to be at least second‐order accurate in space. Numerical results show that the discontinuities, particularly contact discontinuities, can be resolved sharply. The use of AMR allows flow features at disparate scales to be resolved sufficiently. In addition, three‐dimensional shock–bubble interactions are simulated to investigate effects of Mach number on bubble evolution. The flow structures including those peculiar to three‐dimensional bubble are resolved correctly, and some physical phenomena with increasing Mach number are reported. Copyright © 2012 John Wiley & Sons, Ltd.     

Improvements in the reliability and quality of unstructured hybrid mesh generation

This paper presents a reliable and automated approach to the generation of unstructured hybrid grids comprised of tetrahedra, prisms and pyramids for high Reynolds number viscous flow simulations. To enhance robustness, the hybrid mesh generation process starts with the formation of an isotropic tetrahedral grid. Prismatic layers are then added on no‐slip walls fully automatically by obeying user‐specified boundary conditions and three parameters: the number of the layers, an initial layer thickness normal to the walls, and a stretching factor. Topological modifications to the original isotropic tetrahedral elements are prohibited during the layer generation process. The tetrahedral elements near no‐slip walls are shifted inward and the resulting gap between the tetrahedra and the walls is filled up with prismatic elements. To enhance the quality of the prismatic layers around sharp corners, two normals are evaluated for the marching process in these regions. The addition of prismatic elements is locally stopped if negative‐volume elements are created or not enough space is left. An angle‐based smoothing method ensures that the quality of the tetrahedral elements is retained for a reasonable computational cost. The method is demonstrated for two scaled experimental supersonic airplane models designed at the National Aerospace Laboratory of Japan (NAL). Numerical results are compared with wind tunnel experimental data. Copyright © 2004 John Wiley & Sons, Ltd.     

黏性边界层网格自动生成

高雷诺数黏性流动在壁面附近存在边界层,在计算模拟中自动生成可靠且有效的计算网格仍然是计算流体力学存在的瓶颈.三棱柱/四面体混合网格技术在一定程度上缓解了这个困难.然而,对于复杂外形的情况,在边界层内自动高效生成高质量的三棱柱单元仍然十分困难.常用的层推进法在凹凸区域及角点处生成的边界层网格单元质量较差,边界层网格最外层尺寸不均匀.针对这些问题,发展了一种黏性边界层网格自动生成方法,通过顶点周围边的二面角识别物面网格特征确定多生长方向,预估并调整生长高度处理相交情况.同时提出一种三维前沿尺寸调节方式,提高了边界层网格单元的正交性,保证了边界层网格与远场网格尺寸的光滑过渡.通过ONERA M6翼型以及带发动机短舱的DLR-F6翼身组合体等外形的网格生成实例及绕流数值模拟,将计算值与标准实验值进行对比,结果表明:该方法能够自动高效地生成满足数值计算需求的混合网格.     

A gas‐kinetic scheme for the simulation of turbulent flows on unstructured meshes

This paper presents a numerical method for simulating turbulent flows via coupling the Boltzmann BGK equation with Spalart–Allmaras one equation turbulence model. Both the Boltzmann BGK equation and the turbulence model equation are carried out using the finite volume method on unstructured meshes, which is different from previous works on structured grid. The application of the gas‐kinetic scheme is extended to the simulation of turbulent flows with arbitrary geometries. The adaptive mesh refinement technique is also adopted to reduce the computational cost and improve the efficiency of meshes. To organize the unstructured mesh data structure efficiently, a non‐manifold hybrid mesh data structure is extended for polygonal cells. Numerical experiments are performed on incompressible flow over a smooth flat plate and compressible turbulent flows around a NACA 0012 airfoil using unstructured hybrid meshes. These numerical results are found to be in good agreement with experimental data and/or other numerical solutions, demonstrating the applicability of the proposed method to simulate both subsonic and transonic turbulent flows. Copyright © 2016 John Wiley & Sons, Ltd.     

Anisotropic mesh adaptation: towards user‐independent,mesh‐independent and solver‐independent CFD. Part III. Unstructured meshes

The present paper is the third article in a three‐part series on anisotropic mesh adaptation and its application to two‐ and three‐dimensional, structured and unstructured meshes. This third paper concerns the application of the full adaptation methodology to 2‐D unstructured meshes, including all four mesh modification strategies presented in Part I, i.e. refinement/coarsening, edge swapping and node movement. The mesh adaptation procedure is validated through a careful monitoring of a single adaptation step and of the solution–adaptation loop. Independence from the initial mesh and from the flow solver is illustrated. The efficiency of the overall methodology is investigated on relevant laminar and turbulent flow benchmarks. Copyright © 2002 John Wiley & Sons, Ltd.     

A parallel unstructured dynamic mesh adaptation algorithm for 3‐D unsteady flows

An unstructured dynamic mesh adaptation and load balancing algorithm has been developed for the efficient simulation of three‐dimensional unsteady inviscid flows on parallel machines. The numerical scheme was based on a cell‐centred finite‐volume method and the Roe's flux‐difference splitting. Second‐order accuracy was achieved in time by using an implicit Jacobi/Gauss–Seidel iteration. The resolution of time‐dependent solutions was enhanced by adopting an h‐refinement/coarsening algorithm. Parallelization and load balancing were concurrently achieved on the adaptive dynamic meshes for computational speed‐up and efficient memory redistribution. A new tree data structure for boundary faces was developed for the continuous transfer of the communication data across the parallel subdomain boundary. The parallel efficiency was validated by applying the present method to an unsteady shock‐tube problem. The flows around oscillating NACA0012 wing and F‐5 wing were also calculated for the numerical verification of the present dynamic mesh adaptation and load balancing algorithm. Copyright © 2005 John Wiley & Sons, Ltd.     

An adaptive Lagrangian boundary element approach for three‐dimensional transient free‐surface Stokes flow as applied to extrusion,thermoforming, and rheometry

An adaptive (Lagrangian) boundary element approach is proposed for the general three‐dimensional simulation of confined free‐surface Stokes flow. The method is stable as it includes remeshing capabilities of the deforming free surface and thus can handle large deformations. A simple algorithm is developed for mesh refinement of the deforming free‐surface mesh. Smooth transition between large and small elements is achieved without significant degradation of the aspect ratio of the elements in the mesh. Several flow problems are presented to illustrate the utility of the approach, particularly as encountered in polymer processing and rheology. These problems illustrate the transient nature of the flow during the processes of extrusion and thermoforming, the elongation of a fluid sample in an extensional rheometer, and the coating of a sphere. Surface tension effects are also explored. Copyright © 2001 John Wiley & Sons, Ltd.     

A high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows

Adaptive mesh techniques are used widely in the numerical simulations of fluid flows, and the simulation results with high accuracies are obtained by appropriate mesh adaptations. However, gas–liquid two‐phase flows are still difficult to be simulated on adaptive meshes, especially on unstructured adaptive meshes, because the physical phenomena near gas–liquid interfaces are highly complicated and in general, not modeled appropriately on adaptive meshes. In this paper, a high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows is developed and verified/validated. In the unstructured adaptive mesh technique, the PLIC algorithm is employed to simulate interfacial dynamic behaviors and, therefore, the reconstruction method for the interfaces in refined cells is developed, which satisfies the gas and liquid volume conservations and geometrical conservations of interfaces. In addition, the physics‐based consideration is performed on the momentum calculations near interfaces, and the calculation method with gas and liquid momentum conservations is developed. For verification, the slotted‐disk revolution problem is solved. As a result, the unstructured adaptive mesh technique succeeds in reproducing the slotted‐disk shape accurately and well maintaining the shape after one full‐revolution. The dam‐break problem is also simulated and the momentum conservative calculation method succeeds in providing physically appropriate results, which show good agreements with experimental data. Therefore, it is confirmed that the developed unstructured adaptive mesh technique is very efficient to simulate gas–liquid two‐phase flows accurately. Copyright © 2010 John Wiley & Sons, Ltd.     

The direct simulation Monte Carlo method using unstructured adaptive mesh and its application

The implementation of an adaptive mesh‐embedding (h‐refinement) scheme using unstructured grid in two‐dimensional direct simulation Monte Carlo (DSMC) method is reported. In this technique, local isotropic refinement is used to introduce new mesh where the local cell Knudsen number is less than some preset value. This simple scheme, however, has several severe consequences affecting the performance of the DSMC method. Thus, we have applied a technique to remove the hanging node, by introducing the an‐isotropic refinement in the interfacial cells between refined and non‐refined cells. Not only does this remedy increase a negligible amount of work, but it also removes all the difficulties presented in the originals scheme. We have tested the proposed scheme for argon gas in a high‐speed driven cavity flow. The results show an improved flow resolution as compared with that of un‐adaptive mesh. Finally, we have used triangular adaptive mesh to compute a near‐continuum gas flow, a hypersonic flow over a cylinder. The results show fairly good agreement with previous studies. In summary, the proposed simple mesh adaptation is very useful in computing rarefied gas flows, which involve both complicated geometry and highly non‐uniform density variations throughout the flow field. Copyright © 2002 John Wiley & Sons, Ltd.     

Adaptive mesh refinement‐enhanced local DFD method and its application to solve Navier–Stokes equations

Recently, the domain‐free discretization (DFD) method was presented to efficiently solve problems with complex geometries without introducing the coordinate transformation. In order to exploit the high performance of the DFD method, in this paper, the local DFD method with the use of Cartesian mesh is presented, where the physical domain is covered by a Cartesian mesh and the local DFD method is applied for numerical discretization. In order to further improve the efficiency of the solver, the newly developed solution‐based adaptive mesh refinement (AMR) technique is also introduced. The proposed methods are then applied to the simulation of natural convection in concentric annuli between a square outer cylinder and a circular inner cylinder. Numerical experiments show that the present numerical results agree very well with available data in the literature, and AMR‐enhanced local DFD method is an effective tool for the computation of flow problems. Copyright © 2005 John Wiley & Sons, Ltd.     

非结构混合网格在气动热标模模拟中的应用

非结构混合网格消除了结构网格节点的结构性限制,可以较好地处理边界,同时兼顾了粘性边界层模拟的需求,具有灵活性大、对复杂外形适应能力强和生成耗时短等优点,在飞行器气动特性模拟中得到广泛应用.本文针对非结构混合网格的特点,把前期针对非结构混合网格气动力高精度模拟发展改进的梯度计算方法和Roe格式熵修正方法推广应用到气动热流的数值模拟.以典型钝锥标模外形的高超声速绕流为研究对象,开展了不同网格形式和第一层网格不同间距的影响研究.结果 表明,热流计算时,头部物面网格最好采用四边形或四边形交叉剖分得到的三角形网格,物面法向的网格雷诺数取20左右,为热流计算时非结构混合网格的生成提供了指导,同时验证了计算方法的有效性和可靠性.     

覆盖件冲压仿真中的网格形态优化技术

覆盖件冲压仿真计算模型中网格密度分布的合理性与网格单元形态的优劣,对仿真结果的准确性有很大的影响.提出了一种改进的全四边形网格细分方法,使网格的密度分布适于覆盖件冲压分析计算要求,可保证细分后网格的协调性,并将算法推广以处理非结构化四边形网格和三角形四边形混合网格的细分.提出的网格细分策略,有助于提高细分后网格的质量.提出了适用于细分后四边形网格和非结构四边形网格的拓扑形态优化操作,可有效的提高网格模型的形态质量.