High-Resolution Finite Volume Methods on Unstructured Grids for Turbulence and Aeroacoustics

In this paper we focus on the application of a higher-order finite volume method for the resolution of Computational Aeroacoustics problems. In particular, we present the application of a finite volume method based in Moving Least Squares approximations in the context of a hybrid approach for low Mach number flows. In this case, the acoustic and aerodynamic fields can be computed separately. We focus on two kinds of computations: turbulent flow and aeroacoustics in complex geometries. Both fields require very accurate methods to capture the fine features of the flow, small scales in the case of turbulent flows and very low-amplitude acoustic waves in the case of aeroacoustics. On the other hand, the use of unstructured grids is interesting for real engineering applications, but unfortunately, the accuracy and efficiency of the numerical methods developed for unstructured grids is far to reach the performance of those methods developed for structured grids. In this context, we propose the FV-MLS method as a tool for accurate CAA computations on unstructured grids.     

Spectral Difference Method for Unstructured Grids II: Extension to the Euler Equations

An efficient, high-order, conservative method named the spectral difference method has been developed recently for conservation laws on unstructured grids. It combines the best features of structured and unstructured grid methods to achieve high-computational efficiency and geometric flexibility; it utilizes the concept of discontinuous and high-order local representations to achieve conservation and high accuracy; and it is based on the finite-difference formulation for simplicity. The method is easy to implement since it does not involve surface or volume integrals. Universal reconstructions are obtained by distributing solution and flux points in a geometrically similar manner for simplex cells. In this paper, the method is further extended to nonlinear systems of conservation laws, the Euler equations. Accuracy studies are performed to numerically verify the order of accuracy. In order to capture both smooth feature and discontinuities, monotonicity limiters are implemented, and tested for several problems in one and two dimensions. The method is more efficient than the discontinuous Galerkin and spectral volume methods for unstructured grids.     

High Order Finite Difference and Finite Volume Methods for Advection on the Sphere

Numerical schemes used for computational climate modeling and weather prediction are often of second order accuracy. It is well-known that methods of formal order higher than two offer a significant potential gain in computational efficiency. We here present two classes of high order methods for discretization on the surface of a sphere, first finite difference schemes satisfying the summation-by-parts property on the cube sphere grid, secondly finite volume discretizations on unstructured grids with polygonal cells. Furthermore, we also implement the seventh order accurate weighted essentially non-oscillatory (WENO7) scheme for the cube sphere grid. For the finite difference approximation, we prove a stability estimate, derived from projection boundary conditions. For the finite volume method, we develop the implementational details by working in a local coordinate system at each cell. We apply the schemes to compute advection on a sphere, which is a well established test problem. We compare the performance of the methods with respect to accuracy, computational efficiency, and ability to capture discontinuities.     

PyFR: An open source framework for solving advection–diffusion type problems on streaming architectures using the flux reconstruction approach

High-order numerical methods for unstructured grids combine the superior accuracy of high-order spectral or finite difference methods with the geometric flexibility of low-order finite volume or finite element schemes. The Flux Reconstruction (FR) approach unifies various high-order schemes for unstructured grids within a single framework. Additionally, the FR approach exhibits a significant degree of element locality, and is thus able to run efficiently on modern streaming architectures, such as Graphical Processing Units (GPUs). The aforementioned properties of FR mean it offers a promising route to performing affordable, and hence industrially relevant, scale-resolving simulations of hitherto intractable unsteady flows within the vicinity of real-world engineering geometries. In this paper we present PyFR, an open-source Python based framework for solving advection–diffusion type problems on streaming architectures using the FR approach. The framework is designed to solve a range of governing systems on mixed unstructured grids containing various element types. It is also designed to target a range of hardware platforms via use of an in-built domain specific language based on the Mako templating engine. The current release of PyFR is able to solve the compressible Euler and Navier–Stokes equations on grids of quadrilateral and triangular elements in two dimensions, and hexahedral elements in three dimensions, targeting clusters of CPUs, and NVIDIA GPUs. Results are presented for various benchmark flow problems, single-node performance is discussed, and scalability of the code is demonstrated on up to 104 NVIDIA M2090 GPUs. The software is freely available under a 3-Clause New Style BSD license (see www.pyfr.org).     

A Robust Reconstruction for Unstructured WENO Schemes

The weighted essentially non-oscillatory (WENO) schemes are a popular class of high order numerical methods for hyperbolic partial differential equations (PDEs). While WENO schemes on structured meshes are quite mature, the development of finite volume WENO schemes on unstructured meshes is more difficult. A major difficulty is how to design a robust WENO reconstruction procedure to deal with distorted local mesh geometries or degenerate cases when the mesh quality varies for complex domain geometry. In this paper, we combine two different WENO reconstruction approaches to achieve a robust unstructured finite volume WENO reconstruction on complex mesh geometries. Numerical examples including both scalar and system cases are given to demonstrate stability and accuracy of the scheme.     

A unifying grid approach for solving potential flows applicable to structured and unstructured grid configurations

In this study, an efficient numerical method is proposed for unifying the structured and unstructured grid approaches for solving the potential flows. The new method, named as the “alternating cell directions implicit - ACDI”, solves for the structured and unstructured grid configurations equally well. The new method in effect applies a line implicit method similar to the Line Gauss Seidel scheme for complex unstructured grids including mixed type quadrilateral and triangle cells. To this end, designated alternating directions are taken along chains of contiguous cells, i.e. ‘cell directions’, and an ADI-like sweeping is made to update these cells using a Line Gauss Seidel like scheme. The algorithm makes sure that the entire flow field is updated by traversing each cell twice at each time step for unstructured quadrilateral grids that may contain triangular cells. In this study, a cell-centered finite volume formulation of the ACDI method is demonstrated. The solutions are obtained for incompressible potential flows around a circular cylinder and a forward step. The results are compared with the analytical solutions and numerical solutions using the implicit ADI and the explicit Runge-Kutta methods on single-and multi-block structured and unstructured grids. The results demonstrate that the present ACDI method is unconditionally stable, easy to use and has the same computational performance in terms of convergence, accuracy and run times for both the structured and unstructured grids.     

High-order Finite Volume Methods and Multiresolution Reproducing Kernels

This paper presents a review of some of the most successful higher-order numerical schemes for the compressible Navier-Stokes equations on unstructured grids. A suitable candidate scheme would need to be able to handle potentially discontinuous flows, arising from the predominantly hyperbolic character of the equations, and at the same time be well suited for elliptic problems, in order to deal with the viscous terms. Within this context, we explore the performance of Moving Least-Squares (MLS) approximations in the construction of higher order finite volume schemes on unstructured grids. The scope of the application of MLS is threefold: 1) computation of high order derivatives of the field variables for a Godunov-type approach to hyperbolic problems or terms of hyperbolic character, 2) direct reconstruction of the fluxes at cell edges, for elliptic problems or terms of elliptic character, and 3) multiresolution shock detection and selective limiting. The proposed finite volume method is formulated within a continuous spatial representation framework, provided by the MLS approximants, which is “broken” locally (inside each cell) into piecewise polynomial expansions, in order to make use of the specialized finite volume technology for hyperbolic problems. This approach is in contrast with the usual practice in the finite volume literature, which proceeds bottom-up, starting from a piecewise constant spatial representation. Accuracy tests show that the proposed method achieves the expected convergence rates. Representative simulations show that the methodology is applicable to problems of engineering interest, and very competitive when compared to other existing procedures.     

A block LU-SGS implicit unsteady incompressible flow solver on hybrid dynamic grids for 2D external bio-fluid simulations

A hybrid dynamic grid generation technique for two-dimensional (2D) morphing bodies and a block lower-upper symmetric Gauss-Seidel (BLU-SGS) implicit dual-time-stepping method for unsteady incompressible flows are presented for external bio-fluid simulations. To discretize the complicated computational domain around 2D morphing configurations such as fishes and insect/bird wings, the initial grids are generated by a hybrid grid strategy firstly. Body-fitted quadrilateral (quad) grids are generated first near solid bodies. An adaptive Cartesian mesh is then generated to cover the entire computational domain. Cartesian cells which overlap the quad grids are removed from the computational domain, and a gap is produced between the quad grids and the adaptive Cartesian grid. Finally triangular grids are used to fill this gap. During the unsteady movement of morphing bodies, the dynamic grids are generated by a coupling strategy of the interpolation method based on ‘Delaunay graph’ and local remeshing technique. With the motion of moving/morphing bodies, the grids are deformed according to the motion of morphing body boundaries firstly with the interpolation strategy based on ‘Delaunay graph’ proposed by Liu and Qin. Then the quality of deformed grids is checked. If the grids become too skewed, or even intersect each other, the grids are regenerated locally. After the local remeshing, the flow solution is interpolated from the old to the new grid. Based on the hybrid dynamic grid technique, an efficient implicit finite volume solver is set up also to solve the unsteady incompressible flows for external bio-fluid dynamics. The fully implicit equation is solved using a dual-time-stepping approach, coupling with the artificial compressibility method (ACM) for incompressible flows. In order to accelerate the convergence history in each sub-iteration, a block lower-upper symmetric Gauss-Seidel implicit method is introduced also into the solver. The hybrid dynamic grid generator is tested by a group of cases of morphing bodies, while the implicit unsteady solver is validated by typical unsteady incompressible flow case, and the results demonstrate the accuracy and efficiency of present solver. Finally, some applications for fish swimming and insect wing flapping are carried out to demonstrate the ability for 2D external bio-fluid simulations.     

High-Order Flux Correction for Viscous Flows on Arbitrary Unstructured Grids

A novel high-order method, termed flux correction, previously formulated for inviscid flows, is extended to viscous flows on arbitrary triangular grids. The correction method involves the addition of truncation error-canceling terms to the second-order linear Galerkin (node-centered finite volume) scheme to produce a third-order inviscid and fourth-order viscous scheme. The correction requires minimal modification of the underlying second-order scheme. As such, the method retains many of the advantages of traditional finite volume schemes, including robust shock capturing, low algorithmic complexity, and solver efficiency. In addition, we extend the scheme to unsteady flows. Verification and validation studies in two dimensions are presented. Significant improvement in accuracy is observed in all cases, with between 30–70 % increase in computational cost over a second-order finite volume method.     

Numerical simulation of 3D fluid–structure interaction flow using an immersed object method with overlapping grids

The newly developed immersed object method (IOM) [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady incompressible viscous flows around moving rigid bodies using an immersed object method with overlapping grids. J Comput Phys 2005; 207(1): 151–72] is extended for 3D unsteady flow simulation with fluid–structure interaction (FSI), which is made possible by combining it with a parallel unstructured multigrid Navier–Stokes solver using a matrix-free implicit dual time stepping and finite volume method [Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method. In: The second M.I.T. conference on computational fluid and solid mechanics, June 17–20, MIT, Cambridge, MA 02139, USA, 2003; Tai CH, Zhao Y, Liew KM. Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method, Special issue on “Preconditioning methods: algorithms, applications and software environments. Comput Struct 2004; 82(28): 2425–36]. This uniquely combined method is then employed to perform detailed study of 3D unsteady flows with complex FSI. In the IOM, a body force term F is introduced into the momentum equations during the artificial compressibility (AC) sub-iterations so that a desired velocity distribution V₀ can be obtained on and within the object boundary, which needs not coincide with the grid, by adopting the direct forcing method. An object mesh is immersed into the flow domain to define the boundary of the object. The advantage of this is that bodies of almost arbitrary shapes can be added without grid restructuring, a procedure which is often time-consuming and computationally expensive. It has enabled us to perform complex and detailed 3D unsteady blood flow and blood–leaflets interaction in a mechanical heart valve (MHV) under physiological conditions.     

Uniform Second Order Convergence of a Complete Flux Scheme on Unstructured 1D Grids for a Singularly Perturbed Advection–Diffusion Equation and Some Multidimensional Extensions

The accurate and efficient discretization of singularly perturbed advection–diffusion equations on arbitrary 2D and 3D domains remains an open problem. An interesting approach to tackle this problem is the complete flux scheme (CFS) proposed by G. D. Thiart and further investigated by J. ten Thije Boonkkamp. For the CFS, uniform second order convergence has been proven on structured grids. We extend a version of the CFS to unstructured grids for a steady singularly perturbed advection–diffusion equation. By construction, the novel finite volume scheme is nodally exact in 1D for piecewise constant source terms. This property allows to use elegant continuous arguments in order to prove uniform second order convergence on unstructured one-dimensional grids. Numerical results verify the predicted bounds and suggest that by aligning the finite volume grid along the velocity field uniform second order convergence can be obtained in higher space dimensions as well.     

DIMEX Runge–Kutta finite volume methods for multidimensional hyperbolic systems

We propose a class of finite volume methods for the discretization of time-dependent multidimensional hyperbolic systems in divergence form on unstructured grids. We discretize the divergence of the flux function by a cell-centered finite volume method whose spatial accuracy is provided by including into the scheme non-oscillatory piecewise polynomial reconstructions. We assume that the numerical flux function can be decomposed in a convective term and a non-convective term. The convective term, which may be source of numerical stiffness in high-speed flow regions, is treated implicitly, while the non-convective term is always discretized explicitly. To this purpose, we use the diagonally implicit–explicit Runge–Kutta (DIMEX-RK) time-marching formulation. We analyze the structural properties of the matrix operators that result from coupling finite volumes and DIMEX-RK time-stepping schemes by using M-matrix theory. Finally, we show the behavior of these methods by some numerical examples.     

非结构四面体网格上扩散方程的有限体积差分方法

基于二维扩散方程的有限体积方法,构造了三维扩散方程在非结构网格上有限体积差分方法,方法具有高精度和保持通量守恒特性.采取单元中心作为计算节点来减少向量和单元体积的计算量.利用通量守恒条件确定界面中心的函数值,保证了方法的守恒特性.用Lagrange因子插值法更好地适应了非结构网格.采取Bi—CGSTAB方法求解线性代数方程组.计算例子验证方法有效.     

Flux-based method of characteristics for contaminant transport in flowing groundwater

A numerical method for treating advection-dominated contaminant transport in flowing groundwater is described. This method combines advantages of numerical discretizations by finite volume methods (like local mass conservation and the positivity of solutions) and by methods of characteristics (like larger time steps and reduced artificial numerical dispersion). For one-dimensional problems the method can produce equivalent algebraic systems as the finite volume Eulerian-Lagrangian localized adjoint method [13] and the flux-based modified method of characteristics [23] (and some other methods). An extension of the "flux-based methods of characteristics" for complex transport problems on multidimensional unstructured computational grids is the main contribution of this paper. Numerical results are included for a well established test example using a flux-based method of characteristics with aligned finite volumes.     

Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier-Stokes equations

In this paper, we propose a new unified family of arbitrary high order accurate explicit one-step finite volume and discontinuous Galerkin schemes on unstructured triangular and tetrahedral meshes for the solution of the compressible Navier-Stokes equations. This new family of numerical methods has first been proposed in [16] for purely hyperbolic systems and has been called P_NP_M schemes, where N indicates the polynomial degree of the test functions and M is the degree of the polynomials used for flux and source computation. A particular feature of the general P_NP_M schemes is that they contain classical high order accurate finite volume schemes (N=0) as well as standard discontinuous Galerkin methods (M=N) just as special cases, which therefore allows for a direct efficiency comparison.In the application section of this paper we first show numerical convergence results on unstructured meshes obtained for the compressible Navier-Stokes equations with Sutherland’s viscosity law, comparing all third to sixth order accurate P_NP_M schemes with each other. In order to validate the method also in practice we show several classical steady and unsteady CFD applications, such as the laminar boundary layer flow over a flat plate at high Reynolds numbers, flow past a NACA0012 airfoil, the unsteady flows past a circular cylinder and a sphere, the unsteady flows of a compressible mixing layer in two space dimensions and finally we also show applications to supersonic flows with shock Mach numbers up to M_s=10.     

A Grid-Insensitive LDA Method on Triangular Grids Solving the System of Euler Equations

The performance of the classic upwind-type residual distribution (RD) methods on skewed triangular grids are rigorously investigated in this paper. Based on an improved signals distribution, an improved second order RD method based on the LDA approach is proposed to faithfully replicate the flow physics on skewed triangular grids. It will be mathematically and numerically shown that the improved LDA method is found to have minimal accuracy variations when grids are skewed compared to classic RD and cell vertex finite volume methods on scalar equations and system of Euler equations.     

Flux-based level-set method for two-phase flows on unstructured grids

In this paper we deal with the application of the flux-based level set method to moving interface computations on unstructured grids. The focus lies on the overcoming of the known difficulties of level set methods, e.g. accurate computations of important geometric properties, reliable and precise reinitialization of the level set function and the adaption of standard discretization methods to the moving boundary case. The basic building block of our approach is the high-resolution flux-based level set method for general advection equation (Frolkovi? and Mikula in SIAM J Sci Comput 29(2):579–597, 2007, Frolkovi? and Wehner in Comput Vis Sci 12(6):626–650, 2009). We extend this method for the problem of reinitialization of the level set function on unstructured grids by using quadratic interpolation to compute distances for nodes close to the interface. To realize numerical simulation for some applications with moving boundaries, we adapt the approach of ghost fluid method (Gibou and Fedkiw in J Comput Phys 202:577–601, 2005) for unstructured grids. The idea is to describe the development of the moving boundary with a level set formulation while the computational grid remains fixed and the boundary conditions are enforced using some extrapolation. Our main motivation is the numerical solution of two-phase incompressible flow problems. Additionally to previously mentioned steps, we introduce further numerical schemes in the framework of finite volume discretization for the flow. Possible jumps of the pressure and the directional derivative of velocity at the interface are modeled directly within the method using the approach of extended approximation spaces. Besides that, an algorithm for the computations of curvature is considered that exhibits the second order accuracy for some examples. Numerical experiments are provided for the presented methods.     

发动机舱温度场仿真及其影响关系研究

为了全面获得飞行器发动机舱内的电子部件的温度分布,利用有限体积法,采用块结构化网格以及非结构化网格.建立了飞行器发动机舱的有限元模型;应用整体求解耦合问题的有效方法,对此发动机舱内外流场与温度场进行了三维的数值模拟;并对此发动机舱内的温度分布进行了重点数值仿真与分析,而且得出了飞行速度、环境温度等因素的变化对部件温度的影响关系.该模拟结果为发动机舱布局设计提供了参考价值,具有重要的现实意义.     

Compact third-order multidimensional upwind discretization for steady and unsteady flow simulations

We propose a new third-order multidimensional upwind algorithm for the solution of the flow equations on tetrahedral cells unstructured grids. This algorithm has a compact stencil (cell-based computations) and uses a finite element idea when computing the residual over the cell to achieve its third-order (spatial) accuracy. The construction of the new scheme is presented. The asymptotic accuracy for steady or unsteady, inviscid or viscous flow situations is proved using numerical experiments. The new high-order discretization proves to have excellent parallel scalability. Our studies show the advantages of the new compact third-order scheme when compared with the classical second-order multidimensional upwind schemes.     

Explicit Hybrid Time Domain Solver for the Maxwell Equations in 3D

We present an accurate and efficient explicit hybrid solver for Maxwell's equations in time domain. The hybrid solver combines FD-TD with an unstructured finite volume solver. The finite volume solver is a generalization of FD-TD to unstructured grids and it uses a third-order staggered Adams–Bashforth scheme for time discretization. A spatial filter of Laplace type is used by the finite volume solver to enable long simulations without suffering from late time instability problems. The numerical examples demonstrate that the hybrid solver is superior to stand-alone FD-TD in terms of accuracy and efficiency.