Performance study of the multiwavelet discontinuous Galerkin approach for solving the Green-Naghdi equations

This paper presents a multiresolution discontinuous Galerkin (DG) scheme for the adaptive solution of Boussinesq-type equations. The model combines multiwavelet (MW)–based grid adaptation with a DG solver based on the system of fully nonlinear and weakly dispersive Green-Naghdi (GN) equations. The key feature of the adaptation procedure is to conduct a multiresolution analysis using MWs on a hierarchy of nested grids to improve the efficiency of the reference DG scheme on a uniform grid by computing on a locally refined adapted grid. This way, the local resolution level will be determined by manipulating MW coefficients controlled by a single user-defined threshold value. The proposed adaptive multiwavelet DG solver for GN equations is assessed using several benchmark problems related to wave propagation and transformation in nearshore areas. The numerical results demonstrate that the proposed scheme retains the accuracy of the reference scheme, while significantly reducing the computational cost.     

An unstructured/structured multi‐layer hybrid grid method and its application

A multi‐layer hybrid grid method is constructed to simulate complex flow field around 2‐D and 3‐D configuration. The method combines Cartesian grids with structured grids and triangular meshes to provide great flexibility in discretizing a domain. We generate the body‐fitted structured grids near the wall surface and the Cartesian grids for the far field. In addition, we regard the triangular meshes as an adhesive to link each grid part. Coupled with a tree data structure, the Cartesian grid is generated automatically through a cell‐cutting algorithm. The grid merging methodology is discussed, which can smooth hybrid grids and improve the quality of the grids. A cell‐centred finite volume flow solver has been developed in combination with a dual‐time stepping scheme. The flow solver supports arbitrary control volume cells. Both inviscid and viscous flows are computed by solving the Euler and Navier–Stokes equations. The above methods and algorithms have been validated on some test cases. Computed results are presented and compared with experimental data. Copyright © 2006 John Wiley & Sons, Ltd.     

A fast nested multi‐grid viscous flow solver for adaptive Cartesian/Quad grids

A nested multi‐grid solution algorithm has been developed for an adaptive Cartesian/Quad grid viscous flow solver. Body‐fitted adaptive Quad (quadrilateral) grids are generated around solid bodies through ‘surface extrusion’. The Quad grids are then overlapped with an adaptive Cartesian grid. Quadtree data structures are employed to record both the Quad and Cartesian grids. The Cartesian grid is generated through recursive sub‐division of a single root, whereas the Quad grids start from multiple roots—a forest of Quadtrees, representing the coarsest possible Quad grids. Cell‐cutting is performed at the Cartesian/Quad grid interface to merge the Cartesian and Quad grids into a single unstructured grid with arbitrary cell topologies (i.e., arbitrary polygons). Because of the hierarchical nature of the data structure, many levels of coarse grids have already been built in. The coarsening of the unstructured grid is based on the Quadtree data structure through reverse tree traversal. Issues arising from grid coarsening are discussed and solutions are developed. The flow solver is based on a cell‐centered finite volume discretization, Roe's flux splitting, a least‐squares linear reconstruction, and a differentiable limiter developed by Venkatakrishnan in a modified form. A local time stepping scheme is used to handle very small cut cells produced in cell‐cutting. Several cycling strategies, such as the saw‐tooth, W‐ and V‐cycles, have been studies. The V‐cycle has been found to be the most efficient. In general, the multi‐grid solution algorithm has been shown to greatly speed up convergence to steady state—by one to two orders. Copyright © 2000 John Wiley & Sons, Ltd.     

An immersed boundary solver for inviscid compressible flows

In this paper, a simple and efficient immersed boundary (IB) method is developed for the numerical simulation of inviscid compressible Euler equations. We propose a method based on coordinate transformation to calculate the unknowns of ghost points. In the present study, the body‐grid intercept points are used to build a complete bilinear (2‐D)/trilinear (3‐D) interpolation. A third‐order weighted essentially nonoscillation scheme with a new reference smoothness indicator is proposed to improve the accuracy at the extrema and discontinuity region. The dynamic blocked structured adaptive mesh is used to enhance the computational efficiency. The parallel computation with loading balance is applied to save the computational cost for 3‐D problems. Numerical tests show that the present method has second‐order overall spatial accuracy. The double Mach reflection test indicates that the present IB method gives almost identical solution as that of the boundary‐fitted method. The accuracy of the solver is further validated by subsonic and transonic flow past NACA2012 airfoil. Finally, the present IB method with adaptive mesh is validated by simulation of transonic flow past 3‐D ONERA M6 Wing. Global agreement with experimental and other numerical results are obtained.     

动态混合网格生成及隐式非定常计算方法

建立了一种基于动态混合网格的非定常数值计算方法. 混合网格由贴体的四边形网格、外场 的多层次矩形网格和中间的三角形网格构成. 当物体运动时，贴体四边形网格随物体运动而 运动，而外场的矩形网格保持静止，中间的三角形网格随之变形；当物体运动位移较大，导 致三角形网格的质量降低，甚至导致网格相交时，在局部重新生成网格. 新网格上的物理量 由旧网格上的物理量插值而得. 为了提高计算效率，采用了双时间步和子迭代相结合的隐式 有限体积格式计算非定常Navier-Stokes方程. 子迭代采用高效的块LU-SGS方法. 利用该 方法数值模拟了NACA0012振荡翼型的无黏和黏性绕流，得到了与实验和他人计算相当一致 的结果.     

ADAPTIVE SOLUTIONS FOR UNSTEADY LAMINAR FLOWS ON UNSTRUCTURED GRIDS

An adaptive finite volume method for the simulation of time-dependent, viscous flow is presented. The Navier–Stokes equations are discretized by central schemes on unstructured grids and solved by an explicit Runge–Kutta method. The essential topics of the present study are a new concept for a local Runge–Kutta time-stepping scheme, called multisequence Runge–Kutta, which reduces the severe stability restriction in unsteady problems, a common grid generation and adaptation procedure and the application of dynamic grids for capturing moving flow structures. Results are presented for laminar, separated flow around an aerofoil with a flap.     

An Adaptive Grid Eulerian Method for the Computation of Free Surface Flows

A numerical scheme for the prediction of free surface flows is presented and investigated. The method is based on an adaptive grid Eulerian finite-volume method, where non-orthogonal boundary-fitted moving grids are employed to follow the free surface. The underlying flow solver consists in a pressure-correction scheme of SIMPLE type with multigrid acceleration, which is iteratively combined with the moving grid technique. Several numerical examples are considered to illustrate the capabilities of the approach.     

The implementation of an aeronautical CFD flow code onto distributed memory parallel systems

The parallelization of an industrially important in‐house computational fluid dynamics (CFD) code for calculating the airflow over complex aircraft configurations using the Euler or Navier–Stokes equations is presented. The code discussed is the flow solver module of the SAUNA CFD suite. This suite uses a novel grid system that may include block‐structured hexahedral or pyramidal grids, unstructured tetrahedral grids or a hybrid combination of both. To assist in the rapid convergence to a solution, a number of convergence acceleration techniques are employed including implicit residual smoothing and a multigrid full approximation storage scheme (FAS). Key features of the parallelization approach are the use of domain decomposition and encapsulated message passing to enable the execution in parallel using a single programme multiple data (SPMD) paradigm. In the case where a hybrid grid is used, a unified grid partitioning scheme is employed to define the decomposition of the mesh. The parallel code has been tested using both structured and hybrid grids on a number of different distributed memory parallel systems and is now routinely used to perform industrial scale aeronautical simulations. Copyright © 2000 John Wiley & Sons, Ltd.     

A grid deformation technique for unsteady flow computations

A grid deformation technique is presented here based on a transfinite interpolation algorithm applied to the grid displacements. The method, tested using a two‐dimensional flow solver that uses an implicit dual‐time method for the solution of the unsteady Euler equations on deforming grids, is applicable to problems with time varying geometries arising from aeroelasticity and free surface marine problems. The present work is placed into a multi‐block framework and fits into the development of a generally applicable parallel multi‐block flow solver. The effect of grid deformation is examined and comparison with rigidly rotated grids is made for a series of pitching aerofoil test cases selected from the AGARD aeroelastic configurations for the NACA0012 aerofoil. The effect of using a geometric conservation law is also examined. Finally, a demonstration test case for the Williams aerofoil with an oscillating flap is presented, showing the capability of the grid deformation technique. Copyright © 2000 John Wiley & Sons, Ltd.     

A 3D UNSTRUCTURED MESH ADAPTATION ALGORITHM FOR TIME-DEPENDENT SHOCK-DOMINATED PROBLEMS

In this paper we present a tetrahedron-based, h-refinement-type algorithm for the solution of problems in 3D gas dynamics using unstructured mesh adaptation. The mesh adaptation algorithm is coupled to a cell-centred, Riemann problem-based, finite volume scheme of the MUSCL type, employing an approximate Riemann solver. The adaptive scheme is then used to compute the diffraction of shock waves around a box section corner for subsonic and supersonic post-shock flow. In the subsonic case, preliminary measurements of vortex filament speed and vortical Mach number are in broad quantitative agreement with known theoretical results. © 1997 John Wiley & Sons, Ltd.     

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the structured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the solution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.     

Parallel solution of lifting rotors in hover and forward flight

An implicit unsteady, multiblock, multigrid, upwind solver including mesh deformation capability, and structured multiblock grid generator, are presented and applied to lifting rotors in both hover and forward flight. To allow the use of very fine meshes and, hence, better representation of the flow physics, a parallel version of the code has been developed. It is demonstrated that once the grid density is sufficient to capture enough turns of the tip vortices, hover exhibits oscillatory behaviour of the wake, even using a steady formulation. An unsteady simulation is then presented, and detailed analysis of the time‐accurate wake history is performed and compared to theoretical predictions. Forward flight simulations are also presented and, again, grid density effects on the wake formation investigated. Parallel performance of the code using up to 1024 CPU's is also presented. Copyright © 2006 John Wiley & Sons, Ltd.     

Godunov-type solution of the shallow water equations on adaptive unstructured triangular grids

A Godunov-type upwind finite volume solver of the non-linear shallow water equations is described. The shallow water equations are expressed in a hyperbolic conservation law formulation for application to cases where the bed topography is spatially variable. Inviscid fluxes at cell interfaces are computed using Roe's approximate Riemann solver. Second-order accurate spatial calculations of the fluxes are achieved by enhancing the polynomial approximation of the gradients of conserved variables within each cell. Numerical oscillations are curbed by means of a non-linear slope limiter. Time integration is second-order accurate and implicit. The numerical model is based on dynamically adaptive unstructured triangular grids. Test cases include an oblique hydraulic jump, jet-forced flow in a flat-bottomed circular reservoir, wind-induced circulation in a circular basin of non-uniform bed topography and the collapse of a circular dam. The model is found to give accurate results in comparison with published analytical and alternative numerical solutions. Dynamic grid adaptation and the use of a second-order implicit time integration scheme are found to enhance the computational efficiency of the model.     

Acceleration on stretched meshes with line-implicit LU-SGS in parallel implementation

The implicit lower–upper symmetric Gauss–Seidel (LU-SGS) solver is combined with the line-implicit technique to improve convergence on the very anisotropic grids necessary for resolving the boundary layers. The computational fluid dynamics code used is Edge, a Navier–Stokes flow solver for unstructured grids based on a dual grid and edge-based formulation. Multigrid acceleration is applied with the intention to accelerate the convergence to steady state. LU-SGS works in parallel and gives better linear scaling with respect to the number of processors, than the explicit scheme. The ordering techniques investigated have shown that node numbering does influence the convergence and that the orderings from Delaunay and advancing front generation were among the best tested. 2D Reynolds-averaged Navier–Stokes computations have clearly shown the strong efficiency of our novel approach line-implicit LU-SGS which is four times faster than implicit LU-SGS and line-implicit Runge–Kutta. Implicit LU-SGS for Euler and line-implicit LU-SGS for Reynolds-averaged Navier–Stokes are at least twice faster than explicit and line-implicit Runge–Kutta, respectively, for 2D and 3D cases. For 3D Reynolds-averaged Navier–Stokes, multigrid did not accelerate the convergence and therefore may not be needed.     

A composite grid approach for the Euler equations

A zonal grid methodology has been developed for the calculation of compressible fluid flows. The domain subdivision is based on patched grid systems composed of zones or blocks within which a distinct curvilinear grid is generated. The flow simulation is then carried out with a modified scheme based on the Euler finite volume solver of Ni. This scheme uses a distribution procedure that provides an easy and accurate way for the transfer of information from one block to another. This method results in a naturally conservative computation at the interfaces. It is analysed and developed for the treatment of embedded grids with a grid point common to more than four blocks.     

A finite volume Navier-Stokes algorithm for adaptive grids

A compact, finite volume, time-marching scheme for the two-dimensional Navier-Stokes equations of viscous fluid flow is presented. The scheme is designed for unstructured (locally refined) quadrilateral meshes. An earlier inviscid equation (Euler) scheme is employed for the convective terms and the emphasis is on treatment of the viscous terms. An essential feature of the algorithm is that all necessary operations are restricted to within each cell, which is very important when dealing with unstructured grids. Numerical issues which have to be addressed when developing a Navier-Stokes scheme are investigated. These issues are not limited to the particular Navier-Stokes scheme developed in the present work but are general problems. Specifically, the extent of the numerical molecule, which is related to the compactness of the scheme and to its suitability for unstructured grids, is examined. An approach which considers suppression of odd-even mode decoupling of the solution when designing a scheme is presented. In addition, accuracy issues related to grid stretching as well as boundary layer solution contamination due to artificial dissipation are addressed. Although the above issues are investigated with respect to the specific scheme presented, the conclusions are valid for an entire class of finite volume algorithms. The Navier-Stokes solver is validated through test cases which involve comparisons with analytical, numerical and experimental results. The solver is coupled to an adaptive algorithm for high-Reynolds-number aerofoil flow computations.     

基于聚合多重网格方法的跨音速颤振的数值模拟

采用流固耦合方法对跨音速颤振进行了数值模拟。流体方面在非结构网格上用有限体积方法求解了Euler方程;结构方面则求解了后掠机翼典型剖面的结构模态方程。时间推进采用双时间步长：对每一真实时间步,都通过基于聚合多重网格方法的伪时间步推进,对流体和结构方程交替迭代．得到一个稳态的流固耦合的解。文章最后给出了NACA64A010翼型剖面的跨音速颤振边界．与相关文献的计算结果符合良好。     

Octree partitioning of hybrid grids for parallel adaptive viscous flow simulations

A parallel finite volume method for the Navier–Stokes equations with adaptive hybrid prismatic/tetrahedral grids is presented and evaluated in terms of parallel performance. A new method of domain partitioning for complex 3D hybrid meshes is also presented. It is based on orthogonal bisection of a special octree corresponding to the hybrid mesh. The octree is generated automatically and can handle any type of 3D geometry and domain connectivity. One important property of the octree-based partitioning that is exploited is the octree's ability to yield load-balanced partitions that follow the shape of the geometry. This biasing of the octree results in a reduced number of grid elements on the interpartition boundaries and thus fewer data to communicate among processors. Furthermore, the octree-based partitioning gives similar quality of partitions for very different geometries, while requiring minimal user interaction and little computational time. The partitioning method is evaluated in terms of quality of the subdomains as well as execution time. Viscous flow simulations for different geometries are employed to examine the effectiveness of the octree-based partitioning and to test the scalability of parallel execution of the Navier–Stokes solver and hybrid grid adapter on two different parallel systems, the Intel Paragon and the IBM SP2. © 1998 John Wiley & Sons, Ltd.     

A methodology for high performance computation of fully inhomogeneous turbulent flows

A large‐eddy simulation methodology for high performance parallel computation of statistically fully inhomogeneous turbulent flows on structured grids is presented. Strategies and algorithms to improve the memory efficiency as well as the parallel performance of the subgrid‐scale model, the factored scheme, and the Poisson solver on shared‐memory parallel platforms are proposed and evaluated. A novel combination of one‐dimensional red–black/line Gauss–Seidel and two‐dimensional red–black/line Gauss–Seidel methods is shown to provide high efficiency and performance for multigrid relaxation of the Poisson equation. Parallel speedups are measured on various shared‐distributed memory systems. Validations of the code are performed in large‐eddy simulations of turbulent flows through a straight channel and a square duct. Results obtained from the present solver employing a Lagrangian dynamic subgrid‐scale model show good agreements with other available data. The capability of the code for more complex flows is assessed by performing a large‐eddy simulation of the tip‐leakage flow in a linear cascade. Copyright © 2006 John Wiley & Sons, Ltd.     

A new coupled level set and volume-of-fluid method to capture free surface on an overset grid system

We present a new coupled level set and volume-of-fluid (CLSVOF) method for free surface flow simulations on an overset grid system. The coupled method takes advantages of the strengths of the level set (LS) method and the volume-of-fluid (VOF) method, and is superior to either single method. The novelty of the present method lies in that we develop the methodology for an overset grid system of embedding, overlapping and moving structured grids. The new methodology accurately captures interface and greatly preserves mass on an overset grid system by demonstrating the 3D sphere advection test. The method is coupled to a well validated Reynolds-Averaged Navier–Stokes incompressible flow solver. The method is validated with the dam-breaking flow interacting with a 3D obstacle (square structure/circular cylinder) by comparing the numerical results with available experimental and numerical studies. The water impact of a sphere case is further performed to demonstrate the capabilities of the new method on a complicated moving overset grid system.