Transonic drag prediction on a DLR-F6 transport configuration using unstructured grid solvers

A second international AIAA Drag Prediction Workshop (DPW-II) was organized and held in Orlando Florida on June 21-22, 2003. The primary purpose was to investigate the code-to-code uncertainty, address the sensitivity of the drag prediction to grid size and quantify the uncertainty in predicting nacelle/pylon drag increments at a transonic cruise condition. This paper presents an in-depth analysis of the DPW-II computational results from three state-of-the-art unstructured grid Navier-Stokes flow solvers exercised on similar families of tetrahedral grids. The flow solvers are USM3D - a tetrahedral cell-centered upwind solver, FUN3D - a tetrahedral node-centered upwind solver, and NSU3D - a general element node-centered central-differenced solver. Overall, grid refinement did not consistently improve the correlation with experimental data for either the wing/body or the wing/body/nacelle pylon configuration. Although, the range in total drag for the wing/body fine grids was only 5 counts, a code-to-code comparison of surface pressures and surface restricted streamlines indicated that the three solvers were not all converging to the same flow solutions- different shock locations and separation patterns were evident. Similarly, the wing/body/nacelle/pylon solutions did not appear to be converging to the same flow solutions. Although the absolute values of total drag predicted by two of the solvers for the medium and fine grids did not compare well with the experiment, the incremental drag predictions were within ±3 counts of the experimental data. Although, the sources of code-to-code variation in force and moment predictions for the three unstructured grid codes have not yet been identified, the current study reinforces the necessity of applying multiple codes to the same application to assess uncertainty.     

On efficient simulation of non-Newtonian flow in saturated porous media with a multigrid adaptive refinement solver

Flow of non-Newtonian fluid in saturated porous media can be described by the continuity equation and the generalized Darcy law. Here we discuss the efficient solution of the resulting second order nonlinear elliptic equation. The equation is discretized by the finite volume method on a cell-centered grid. Local adaptive refinement of the grid is introduced in order to reduce the number of unknowns. We develop a special implementation, that allows us to perform unstructured local refinement in conjunction with the finite volume discretization. Two residual based error indicators are exploited in the adaptive refinement criterion. Second order accurate discretization of the fluxes on the interfaces between refined and non-refined subdomains, as well as on the boundaries with Dirichlet boundary condition, are presented here as an essential part of an accurate and efficient algorithm. A nonlinear full approximation storage multigrid algorithm is developed especially for the above described composite (coarse plus locally refined) grid approach. In particular, second order approximation of the fluxes around interfaces is a result of a quadratic approximation of slave nodes in the multigrid-adaptive refinement (MG-AR) algorithm. Results from numerical solution of various academic and practice-induced problems are presented and the performance of the solver is discussed.     

Parallel computation of unsteady three-dimensional incompressible viscous flow using an unstructured multigrid method

The development and validation of a parallel unstructured tetrahedral non-nested multigrid (MG) method for simulation of unsteady 3D incompressible viscous flow is presented. The Navier-Stokes solver is based on the artificial compressibility method (ACM) and a higher-order characteristics-based finite-volume scheme on unstructured MG. Unsteady flow is calculated with an implicit dual time stepping scheme. The parallelization of the solver is achieved by a MG domain decomposition approach (MG-DD), using the Single Program Multiple Data (SPMD) programming paradigm. The Message-Passing Interface (MPI) Library is used for communication of data and loop arrays are decomposed using the OpenMP standard. The parallel codes using single grid and MG are used to simulate steady and unsteady incompressible viscous flows for a 3D lid-driven cavity flow for validation and performance evaluation purposes. The speedups and efficiencies obtained by both the parallel single grid and MG solvers are reasonably good for all test cases, using up to 32 processors on the SGI Origin 3400. The parallel results obtained agree well with those of serial solvers and with numerical solutions obtained by other researchers, as well as experimental measurements.     

Flow simulations in arbitrarily complex cardiovascular anatomies – An unstructured Cartesian grid approach

Image guided computational fluid dynamics is attracting increasing attention as a tool for refining in vivo flow measurements or predicting the outcome of different surgical scenarios. Sharp interface Cartesian/Immersed-Boundary methods constitute an attractive option for handling complex in vivo geometries but their capability to carry out fine-mesh simulations in the branching, multi-vessel configurations typically encountered in cardiovascular anatomies or pulmonary airways has yet to be demonstrated. A major computational challenge stems from the fact that when such a complex geometry is immersed in a rectangular Cartesian box the excessively large number of grid nodes in the exterior of the flow domain imposes an unnecessary burden on both memory and computational overhead of the Cartesian solver without enhancing the numerical resolution in the region of interest. For many anatomies, this added burden could be large enough to render comprehensive mesh refinement studies impossible. To remedy this situation, we recast the original structured Cartesian formulation of Gilmanov and Sotiropoulos [Gilmanov A, Sotiropoulos F. A hybrid Cartesian/immersed boundary method for simulating flows with 3D, geometrically complex, moving bodies. J Comput Phys 2005;207(2):457–92] into an unstructured Cartesian grid layout. This simple yet powerful approach retains the simplicity and computational efficiency of a Cartesian grid solver, while drastically reducing its memory footprint. The method is applied to carry out systematic mesh refinement studies for several internal flow problems ranging in complexity from flow in a 90° pipe bend to flow in an actual, patient-specific anatomy reconstructed from magnetic resonance images. Finally, we tackle the challenging clinical scenario of a single-ventricle patient with severe arterio-venous malformations, seeking to provide a fluid dynamics prospective on a clinical problem and suggestions for procedure improvements. Results from these simulations demonstrate very complex cardiovascular flow dynamics and underscore the need for high-resolution simulations prior to drawing any clinical recommendations.     

Semi-Unstructured Grids

We present a novel automatic grid generator for the finite element discretization of partial differential equations in 3D. The grids constructed by this grid generator are composed of a pure tensor product grid in the interior of the domain and an unstructured grid which is only contained in boundary cells. The unstructured component consists of tetrahedra, each of which satisfies a maximal interior angle condition. By suitable constructing the boundary cells, the number of types of boundary subcells is reduced to 12 types. Since this grid generator constructs large structured grids in the interior and small unstructured grids near the boundary, the resulting semi-unstructured grids have similar properties as structured tensor product grids. Some appealing properties of this method are computational efficiency and natural construction of coarse grids for multilevel algorithms. Numerical results and an analysis of the discretization error are presented. Received July 17, 2000; revised October 27, 2000     

非结构粘性直角网格并行算法建立与应用

开展了基于粘性直角非结构网格的并行CFD解算软件的开发研究,工作分两部分：网格分区实现和解算器并行实施,文章介绍了关于CFD解算器并行的实施情况,给出了并行过程中的操作流程,并对一些关键问题进行了讨论。并行计算结果表明项目所采用的并行途径和方法有效,计算结果可靠。     

Incompressible flow computations over moving boundary using a novel upwind method

In this paper a novel method for simulating unsteady incompressible viscous flow over a moving boundary is described. The numerical model is based on a 2D Navier–Stokes incompressible flow in artificial compressibility formulation with Arbitrary Lagrangian Eulerian approach for moving grid and dual time stepping approach for time accurate discretization. A higher order unstructured finite volume scheme, based on a Harten Lax and van Leer with Contact (HLLC) type Riemann solver for convective fluxes, developed for steady incompressible flow in artificial compressibility formulation by Mandal and Iyer (AIAA paper 2009-3541), is extended to solve unsteady flows over moving boundary. Viscous fluxes are discretized in a central differencing manner based on Coirier’s diamond path. An algorithm based on interpolation with radial basis functions is used for grid movements. The present numerical scheme is validated for an unsteady channel flow with a moving indentation. The present numerical results are found to agree well with experimental results reported in literature.     

One- and two-equation turbulence models for the prediction of complex cascade flows using unstructured grids

One- and two-equation, low-Reynolds eddy-viscosity turbulence models are employed in the context of a primitive variable, finite volume, Navier-Stokes solver for unstructured grids. Through the study of the complex flow in a controlled-diffusion compressor cascade at off-design conditions, the ability of the models under consideration to predict the laminar separation bubble close to the leading edge and the boundary layer development is investigated. In order to control the unphysical growth of turbulent kinetic energy near the leading edge stagnation point, appropriate modifications to the conventional models are employed and tested. All of them improve the leading edge flow patterns and significantly affect the size of the predicted laminar separation bubble. The use of an adequately refined mesh around the airfoil, that is formed by triangles placed in a quasi-structured way, allows for the generation of grid elements of moderate aspect ratios. This helps to readily overcome any relevant problems of accuracy; a second-order upwind scheme without flux limiters or least squares approximations is successfully employed for the gradients. The test case includes quasi-3D effects by considering the streamtube thickness variation in the governing equations.     

Automatic mesh refinement and data structure for multigrid finite elements techniques

A general algorithm for locally refining any conforming triangulation to generate a new conforming one is presented. The proposed algorithm ensures that all angles in subsequent refined triangulations are greater than, or equal to, half the smallest angle in the original triangulation, the shape regularity of all triangles is maintained and the transition between small and large triangles is smooth. The generated triangulations are nested, so it is possible to implement the approach with adaptive and/or multigrid techniques. A complete algorithm for solving two-dimensional elliptic boundary value problems adaptively by multigrid is presented. The development and implementation of the main parts of this algorithm; automatic mesh generator, a posteriori error estimator, refinement strategy and the multigrid solver are presented in some detail. An appropriate data structure is developed to meet the excess data required for the generation process also to keep track of different grid levels. By the aid of this data structure, it becomes easy to design simple algorithms to store only the non-zero elements of stiffness matrices for different grids and to design a very simple multigrid transfer operator. Numerical examples are presented to show the generated grid sequence for two different boundary value problems.     

A folding/unfolding algorithm for the construction of semi-structured layers in hybrid grid generation

A method is presented for generating semi-structured layers in hybrid grids suitable for high Reynolds number flow simulations. It can produce superior quality grids around convex and concave ridges of the domain compared to a standard prismatic or advancing layers generator. In a first step, hybrid quadrilateral/triangular grids are employed to cluster the surface grid towards sharp corners and ridges. Starting from this surface grid as an initial front for the volume grid generation, a prismatic and hexahedral layers generation technique is used, containing degenerate faces to add or remove elements in convex or concave areas of the geometry respectively. The method can automatically create structured-like grids in concave gaps and better discretize the space around convex ridges to improve overall accuracy. The final grid remains conforming to not complicate the solver and post-processing requirements. Several examples illustrate the application of the method.     

Computation of complex turbulent flow using matrix-free implicit dual time-stepping scheme and LRN turbulence model on unstructured grids

In this study, a matrix-free implicit dual time-stepping method has been developed. It is implemented, together with a low-Reynolds-number q-ω turbulence model, in a high-order upwind finite-volume solver on unstructured grids. Semi-implicit treatment of the source terms of the q and ω equations is also introduced to further stabilize the numerical solution. It has been found that these techniques provide strong stabilization in the computation of a supersonic flow with complex shock-boundary-layer interactions in a channel with a backward-facing step. The proposed method has a low-memory overhead, similar to an explicit scheme, while it shows good stability and computational efficiency as an implicit scheme. The method developed has been validated by comparing the computed results with the corresponding experimental measurements and other calculated results, which shows good agreement. Research is being done to extend the method to calculate unsteady turbulent flows.     

An upwind discretization scheme for the finite volume lattice Boltzmann method

The fact that the classic lattice Boltzmann method is restricted to Cartesian Grids has inspired several researchers to apply Finite Volume [Nannelli F, Succi S. The lattice Boltzmann equation on irregular lattices. J Stat Phys 1992;68:401–7; Peng G, Xi H, Duncan C, Chou SH. Finite volume scheme for the lattice Boltzmann method on unstructured meshes. Phys Rev E 1999;59:4675–82; Chen H. Volumetric formulation of the lattice Boltzmann method for fluid dynamics: basic concept. Phys Rev E 1998;58:3955–63] or Finite Element [Lee T, Lin CL. A characteristic Galerkin method for discrete Boltzmann equation. J Comp Phys 2001;171:336–56; Shi X, Lin J, Yu Z. Discontinuous Galerkin spectral element lattice Boltzmann method on triangular element. Int J Numer Methods Fluids 2003;42:1249–61] methods to the Discrete Boltzmann equation. The finite volume method proposed by Peng et al. works on unstructured grids, thus allowing an increased geometrical flexibility. However, the method suffers from substantial numerical instability compared to the standard LBE models. The computational efficiency of the scheme is not competitive with standard methods.We propose an alternative way of discretizing the convection operator using an upwind scheme, as opposed to the central scheme described by Peng et al. We apply our method to some test problems in two spatial dimensions to demonstrate the improved stability of the new scheme and the significant improvement in computational efficiency. Comparisons with a lattice Boltzmann solver working on a hierarchical grid were done and we found that currently finite volume methods for the discrete Boltzmann equation are not yet competitive as stand alone fluid solvers.     

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.     

Use of local grid refinement and a Galerkin technique to study secondary migration in fractured and faulted regions

A simulator, which includes a full three dimensional, multi-component and three phase model for secondary oil migration is used to study flow through fractured and faulted regions. A control volume discretization is used to discretize the strongly coupled system of partial differential equations the model creates. Non-regular and non-matching local grid refinement (LGR) is used in critical areas to ensure the required accuracy. Two solution methods for solving the composite problem LGR create are studied. The simplest one uses the GMRES solver directly on the composite problem. The second method splits the composite problem into one coarse problem and one problem for each refined area. This method is a two level multigrid method which is based on a Galerkin technique. The CPU time and the number of iteration needed for convergence in both methods are presented and compared. Received: 23 February 1999 / Accepted: 17 June 1999     

A FV-TD electromagnetic solver using adaptive Cartesian grids

A second-order finite-volume (FV) method has been developed to solve the time-domain (TD) Maxwell equations, which govern the dynamics of electromagnetic waves. The computational electromagnetic (CEM) solver is capable of handling arbitrary grids, including structured, unstructured, and adaptive Cartesian grids, which are topologically arbitrary. It is argued in this paper that the adaptive Cartesian grid is better than a tetrahedral grid for complex geometries considering both efficiency and accuracy. A cell-wise linear reconstruction scheme is employed to achieve second-order spatial accuracy. Second-order time accuracy is obtained through a two-step Runge-Kutta scheme. Issues on automatic adaptive Cartesian grid generation such as cell-cutting and cell-merging are discussed. A multi-dimensional characteristic absorbing boundary condition (MDC-ABC) is developed at the truncated far-field boundary to reduce reflected waves from this artificial boundary. The CEM solver is demonstrated with several test cases with analytical solutions.     

非结构网格的并行多重网格解算器

多重网格方法作为非结构网格的高效解算器,其串行与并行实现在时空上都具有优良特性.以控制方程离散过程为切入点,说明非结构网格在并行数值模拟的流程,指出多重网格方法主要用于求解时间推进格式产生的大规模代数系统方程,简述了算法实现的基本结构,分析了其高效性原理;其次,综述性地概括了几何多重网格与代数多种网格研究动态,并对其并行化的热点问题进行重点论述.同时,针对非结构网格的实际应用,总结了多重网格解算器采用的光滑算子;随后列举了非结构网格应用的部分开源项目软件,并简要说明了其应用功能;最后,指出并行多重网格解算器在非结构网格应用中的若干关键问题和未来的研究方向.     

An algorithm for ideal multigrid convergence for the steady Euler equations

A fast multigrid solver for the steady incompressible Euler equations is presented. Unlike time-marching schemes this approach uses relaxation of the steady equations. Application of this method results in a discretization that correctly distinguishes between the advection and elliptic parts of the operator, allowing efficient smoothers to be constructed. Solvers for both unstructured triangular grids and structured quadrilateral grids have been written. Flows in two-dimensional channels and over airfoils have been computed. Using Gauss–Seidel relaxation with the grid vertices ordered in the flow direction, ideal multigrid convergence rates of nearly one order-of-magnitude residual reduction per multigrid cycle are achieved, independent of the grid spacing. This approach also may be applied to the compressible Euler equations and the incompressible Navier–Stokes equations.     

Computing low Mach number flows by parallel adaptive multigrid

A method to simulate compressible flows at all speed on unstructured grids is presented. Special effort is set on low Mach number flows both in two and three dimensions. Weakly compressible flows can be treated by applying an asymptotic analysis to the equations. The behaviour of a multigrid method solving the linearized system is shown. Furthermore adaptive grid refinement and parallelization is used.This work was supported by the Deutsche Forschungsgemeinschaft, Schwertpunktprogramm ANuME.     

Time-accurate Navier-Stokes simulation of vortex convection using an unstructured dynamic mesh procedure

A two-dimensional Navier-Stokes flow solver is developed for the simulation of unsteady flows on unstructured adaptive meshes. The solver is based on a second-order accurate implicit time integration using a point Gauss-Seidel relaxation scheme and a dual time-step subiteration. A vertex-centered, finite-volume discretization is used in conjunction with Roe’s flux-difference splitting. The Spalart-Allmaras one equation model is employed for the simulation of turbulence. An unsteady solution-adaptive dynamic mesh scheme is used by adding and deleting mesh points to take account of spatial and temporal variations of the flowfield. Unsteady viscous flow for a traveling vortex in a free stream is simulated to validate the accuracy of the dynamic mesh adaptation procedure. Flow around a circular cylinder and two blade-vortex interaction problems are investigated for demonstration of the present method. Computed results show good agreement with existing experimental and computational results. It was found that unsteady time-accurate viscous flows can be accurately simulated using the present unstructured dynamic mesh adaptation procedure.