An implicit mixed finite-volume–finite-element method for solving 3D turbulent compressible flows

The development of new aeronautic projects require accurate and efficient simulations of compressible flows in complex geometries. It is well known that most flows of interest are at least locally turbulent and that the modelling of this turbulence is critical for the reliability of the computations. A turbulence closure model which is both cheap and reasonably accurate is an essential part of a compressible code. An implicit algorithm to solve the 2D and 3D compressible Navier–Stokes equations on unstructured triangular/tetrahedral grids has been extended to turbulent flows. This numerical scheme is based on second-order finite element–finite volume discretization: the diffusive and source terms of the Navier–Stokes equations are computed using a finite element method, while the other terms are computed with a finite volume method. Finite volume cells are built around each node by means of the medians. The convective fluxes are evaluated with the approximate Riemann solver of Roe coupled with the van Albada limiter. The standard k–ϵ model has been introduced to take into account turbulence. Implicit integration schemes with efficient numerical methods (CGS, GMRES and various preconditioning techniques) have also been implemented. Our interest is to present the whole method and to demonstrate its limitations on some well-known test cases in three-dimensional geometries. © 1997 John Wiley & Sons, Ltd.     

Computation of Compressible Flows using a Two-equation Turbulence Model on Unstructured Grids

This paper presents a numerical method for solving compressible turbulent flows using a k - l turbulence model on unstructured meshes. The flow equations and turbulence equations are solved in a loosely coupled manner. The flow equations are advanced in time using a multi-stage Runge-Kutta time stepping scheme, while the turbulence equations are advanced using a multi-stage point-implicit scheme. The positivity of turbulence variables is achieved using a simple change of dependent variables. The developed method is used to compute a variety of turbulent flow problems. The results obtained are in good agreement with theoretical and experimental data, indicating that the present method provides a viable and robust algorithm for computing turbulent flows on unstructured meshes.     

An efficient implicit unstructured finite volume solver for generalised Newtonian fluids

An implicit finite volume solver is developed for the steady-state solution of generalised Newtonian fluids on unstructured meshes in 2D. The pseudo-compressibility technique is employed to couple the continuity and momentum equations by transforming the governing equations into a hyperbolic system. A second-order accurate spatial discretisation is provided by performing a least-squares gradient reconstruction within each control volume of unstructured meshes. A central flux function is used for the convective terms and a solution jump term is added to the averaged component for the viscous terms. Global implicit time-stepping using successive evolution–relaxation is utilised to accelerate the convergence to steady-state solutions. The performance of our flow solver is examined for power-law and Carreau–Yasuda non-Newtonian fluids in different geometries. The effects of model parameters and Reynolds number are studied on the convergence rate and flow features. Our results verify second-order accuracy of the discretisation and also fast and efficient convergence to the steady-state solution for a wide range of flow variables.     

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.     

Turbulent flow calculations using unstructured and adaptive meshes

A method of efficiently computing turbulent compressible flow over complex two-dimensional configurations is presented. The method makes use of fully unstructured meshes throughout the entire flow field, thus enabling the treatment of arbitrarily complex geometries and the use of adaptive meshing techniques throughout both viscous and inviscid regions of the flow field. Mesh generation is based on a locally mapped Delaunay technique in order to generate unstructured meshes with highly stretched elements in the viscous regions. The flow equations are discretized using a finite element Navier-Stokes solver, and rapid convergence to steady state is achieved using an unstructured multigrid algorithm. Turbulence modelling is performed using an inexpensive algebraic model, implemented for use on unstructured and adaptive meshes. Compressible turbulent flow solutions about multiple-element aerofoil geometries are computed and compared with experimental data.     

A Hybrid Domain Decomposition and Multigrid Method for the Acceleration of Compressible Viscous Flow Calculations on Unstructured Triangular Meshes

This paper is concerned with the formulation and the evaluation of a hybrid solution method that makes use of domain decomposition and multigrid principles for the calculation of two-dimensional compressible viscous flows on unstructured triangular meshes. More precisely, a non-overlapping additive domain decomposition method is used to coordinate concurrent subdomain solutions with a multigrid method. This hybrid method is developed in the context of a flow solver for the Navier-Stokes equations which is based on a combined finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi-discrete equations is performed using a linearized backward Euler implicit scheme. As a result, each pseudo time step requires the solution of a sparse linear system. In this study, a non-overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. Algebraically, the Schwarz algorithm is equivalent to a Jacobi iteration on a linear system whose matrix has a block structure. A substructuring technique can be applied to this matrix in order to obtain a fully implicit scheme in terms of interface unknowns. In the present approach, the interface unknowns are numerical fluxes. The interface system is solved by means of a full GMRES method. Here, the local system solves that are induced by matrix-vector products with the interface operator, are performed using a multigrid by volume agglomeration method. The resulting hybrid domain decomposition and multigrid solver is applied to the computation of several steady flows around a geometry of NACA0012 airfoil.     

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.     

A Fast,Matrix-free Implicit Method for Computing Low Mach Number Flows on Unstructured Grids

A fast, matrix-free implicit method has been developed to solve low Mach number flow problems on unstructured grids. The preconditioned compressible Euler and Navier-Stokes equations are integrated in time using a linearized implicit scheme. A newly developed fast, matrix-free implicit method, GMRES + LU?SGS, is then applied to solve the resultant system of linear equations. A variety of computations has been made for a wide range of flow conditions, for both in viscid and viscous flows, in both 2D and 3D to validate the developed method and to evaluate the effectiveness of the GMRES + LU?SGS method. The numerical results obtained indicate that the use of the GMRES + LU?SGS method leads to a significant increase in performance over the LU?SGS method, while maintaining memory requirements similar to its explicit counterpart. An overall speedup factor from one to more than two order of magnitude for all test cases in comparison with the explicit method is demonstrated.     

Finite element solution of three‐dimensional turbulent flows applied to mold‐filling problems

This paper presents a finite element solution algorithm for three‐dimensional isothermal turbulent flows for mold‐filling applications. The problems of interest present unusual challenges for both the physical modelling and the solution algorithm. High‐Reynolds number transient turbulent flows with free surfaces have to be computed on complex three‐dimensional geometries. In this work, a segregated algorithm is used to solve the Navier–Stokes, turbulence and front‐tracking equations. The streamline–upwind/Petrov–Galerkin method is used to obtain stable solutions to convection‐dominated problems. Turbulence is modelled using either a one‐equation turbulence model or the κ–ε two‐equation model with wall functions. Turbulence equations are solved for the natural logarithm of the turbulence variables. The change of dependent variables allows for a robust solution algorithm and good predictions even on coarse meshes. This is very important in the case of large three‐dimensional applications for which highly refined meshes result in untreatable large numbers of elements. The position of the flow front in the mold cavity is computed using a level set approach. Finally, equations are integrated in time using an implicit Euler scheme. The methodology presents the robustness and cost effectiveness needed to tackle complex industrial applications. Copyright © 2000 John Wiley & Sons, Ltd.     

An efficient edge based data structure for the compressible Reynolds-averaged Navier–Stokes equations on hybrid unstructured meshes

An efficient edge based data structure has been developed in order to implement an unstructured vertex based finite volume algorithm for the Reynolds-averaged Navier–Stokes equations on hybrid meshes. In the present approach, the data structure is tailored to meet the requirements of the vertex based algorithm by considering data access patterns and cache efficiency. The required data are packed and allocated in a way that they are close to each other in the physical memory. Therefore, the proposed data structure increases cache performance and improves computation time. As a result, the explicit flow solver indicates a significant speed up compared to other open-source solvers in terms of CPU time. A fully implicit version has also been implemented based on the PETSc library in order to improve the robustness of the algorithm. The resulting algebraic equations due to the compressible Navier–Stokes and the one equation Spalart–Allmaras turbulence equations are solved in a monolithic manner using the restricted additive Schwarz preconditioner combined with the FGMRES Krylov subspace algorithm. In order to further improve the computational accuracy, the multiscale metric based anisotropic mesh refinement library PyAMG is used for mesh adaptation. The numerical algorithm is validated for the classical benchmark problems such as the transonic turbulent flow around a supercritical RAE2822 airfoil and DLR-F6 wing-body-nacelle-pylon configuration. The efficiency of the data structure is demonstrated by achieving up to an order of magnitude speed up in CPU times.     

A semi-coarsening strategy for unstructured multigrid based on agglomeration

Extending multigrid concepts to the calculation of complex compressible flow is usually not straightforward. This is especially true when non-embedded grid hierarchies or volume agglomeration strategies are used to construct a gradation of unstructured grids. In this work, a multigrid method for solving second-order PDE's on stretched unstructured triangulations is studied. The finite volume agglomeration multigrid technique originally developed for solving the Euler equations is used (M.-H. Lallemand and A. Dervieux, in Multigrid Methods, Theory, Applications and Supercomputing, Marcel Dekker, 337–363 (1988)). First, a directional semi-coarsening strategy based on Poisson's equation is proposed. The second-order derivatives are approximated on each level by introducing a correction factor adapted to the semi-coarsening strategy. Then, this method is applied to solve the Poisson equation. It is extended to the 2D Reynolds-averaged Navier–Stokes equations with appropriate boundary treatment for low-Reynolds number turbulent flows. © 1998 John Wiley & Sons, Ltd.     

Flexible Unstructured Gridding for 2D Reservoir Coarse Scale Modeling using Fine Scale Permeability Filtering

Accurate up-scaling is an essential part of creating a valid reservoir coarse scale dynamic model. In this article, unstructured discretization of spatial domain is accompanied by numerical permeability up-scaling in order to construct an accurate coarse scale model. A new technique for generating a course scale triangular mesh is presented in which the density of elements in key flow regions is kept high to capture accuracy. The fine scale permeability map is investigated using image processing techniques, especially steerable filters, and the results are converted into a high-resolution element size map. This element size map will be refined by the integration of other important factors such as well-position effects and used to construct a coarse triangular mesh. The combination of flux-continuous pressure approximation and mass conservative, total variation diminishing finite volume schemes have been considered to solve two phase flow equations on the control volume finite element mesh. Fine scale simulations results are compared with the coarse scale ones for a series of water flooding examples to investigate the efficiency and accuracy of the presented gridding methodology. This method is developed for 2D cases, but can be easily extended to 3D problems.     

A parallel implicit domain decomposition algorithm for the large eddy simulation of incompressible turbulent flows on 3D unstructured meshes

We present a parallel fully implicit algorithm for the large eddy simulation (LES) of incompressible turbulent flows on unstructured meshes in three dimensions. The LES governing equations are discretized by a stabilized Galerkin finite element method in space and an implicit second-order backward differentiation scheme in time. To efficiently solve the resulting large nonlinear systems, we present a highly parallel Newton-Krylov-Schwarz algorithm based on domain decomposition techniques. Analytic Jacobian is applied in order to obtain the best achievable performance. Two benchmark problems of lid-driven cavity and flow passing a square cylinder are employed to validate the proposed algorithm. We then apply the algorithm to the LES of turbulent flows passing a full-size high-speed train with realistic geometry and operating conditions. The numerical results show that the algorithm is both accurate and efficient and exhibits a good scalability and parallel efficiency with tens of millions of degrees of freedom on a computer with up to 4096 processors. To understand the numerical behavior of the proposed fully implicit scheme, we study several important issues, including the choices of linear solvers, the overlapping size of the subdomains, and, especially, the accuracy of the Jacobian matrix. The results show that an exact Jacobian is necessary for the efficiency and the robustness of the proposed LES solver.     

An embedded boundary framework for compressible turbulent flow and fluid–structure computations on structured and unstructured grids

The finite volume method with exact two‐phase Riemann problems (FIVER) is a two‐faceted computational method for compressible multi‐material (fluid–fluid, fluid–structure, and multi‐fluid–structure) problems characterized by large density jumps, and/or highly nonlinear structural motions and deformations. For compressible multi‐phase flow problems, FIVER is a Godunov‐type discretization scheme characterized by the construction and solution at the material interfaces of local, exact, two‐phase Riemann problems. For compressible fluid–structure interaction (FSI) problems, it is an embedded boundary method for computational fluid dynamics (CFD) capable of handling large structural deformations and topological changes. Originally developed for inviscid multi‐material computations on nonbody‐fitted structured and unstructured grids, FIVER is extended in this paper to laminar and turbulent viscous flow and FSI problems. To this effect, it is equipped with carefully designed extrapolation schemes for populating the ghost fluid values needed for the construction, in the vicinity of the fluid–structure interface, of second‐order spatial approximations of the viscous fluxes and source terms associated with Reynolds averaged Navier–Stokes (RANS)‐based turbulence models and large eddy simulation (LES). Two support algorithms, which pertain to the application of any embedded boundary method for CFD to the robust, accurate, and fast solution of FSI problems, are also presented in this paper. The first one focuses on the fast computation of the time‐dependent distance to the wall because it is required by many RANS‐based turbulence models. The second algorithm addresses the robust and accurate computation of the flow‐induced forces and moments on embedded discrete surfaces, and their finite element representations when these surfaces are flexible. Equipped with these two auxiliary algorithms, the extension of FIVER to viscous flow and FSI problems is first verified with the LES of a turbulent flow past an immobile prolate spheroid, and the computation of a series of unsteady laminar flows past two counter‐rotating cylinders. Then, its potential for the solution of complex, turbulent, and flexible FSI problems is also demonstrated with the simulation, using the Spalart–Allmaras turbulence model, of the vertical tail buffeting of an F/A‐18 aircraft configuration and the comparison of the obtained numerical results with flight test data. Copyright © 2014 John Wiley & Sons, Ltd.     

A PRESSURE CORRECTION METHOD FOR UNSTRUCTURED MESHES WITH ARBITRARY CONTROL VOLUMES

A pressure correction procedure for general unstructured meshes is presented. It is a cell-centred, collocated finite volume method and the pressure–velocity coupling is treated using SIMPLEC. The cells can have an arbitrary number of grid points (cell vertices). In the present study the number of faces on the cells varies between three and six. The discretized equations are solved using either a symmetric Gauss–Seidel solver or a conjugate gradient solver with a preconditioner. The method is applied to three two-dimensional test cases in which the flow is incompressible and laminar. The extension to three dimensions as well as to turbulent flow using transport models is straightforward. It can also be extended to handle compressible flow.     

FV/MC混合算法求解轴对称钝体后湍流流场

介绍一种有限容积／Monte Carlo结合求解湍流流场的相容的混合算法．有限容积法求解Reynolds平均的动量方程和能量方程，Monte Carlo方法求解模化的脉动速度—频率—标量联合的PDF方程．将该算法发展到无结构网格，探讨了在无结构网格中实现两种方法的耦合，包括颗粒定位，颗粒场和平均场之间数据交换等问题．并以二维轴对称钝体后湍流流场作为算例，比较了计算结果与实验结果．     

亚、跨、超音速及不可压流动的数值分析方法的研究

为了对亚、跨、超音速及不可压无粘流动进行数值模拟,将LU－SGS方法与预处理方法结合,给出了PLU－SGS方法。方程离散基于有限体积法,采用高阶精度AUSMPW格式。方程求解采用了特征边界条件。通过典型算例的数值试验对比分析,表明PLU－SGS方法可以有效地对亚、跨、超音速及不可压流动进行数值模拟,并具有较高的计算精度和收敛速度。     

An upwind scheme for the three-dimensional boundary layer equations

The development of a calculation method to solve the compressible, three-dimensional, turbulent boundary layer equations is described. An implicit finite difference solution procedure is adopted involving local upwinding of convective transport terms. A consistent approach to discretization and linearization is taken by casting all equations in a similar form. The implementation of algebraic, one-equation and two-equation turbulence models is described. An initial validation of the method is made by comparing prediction with measurements in two quasi-three-dimensional boundary layer flows. Some of the more obvious deficiencies in current turbulence-modelling standards for three-dimensional flows are discussed.     

Numerical solution of 2D and 3D turbulent internal flow problems

The paper describes a method for solving numerically two-dimensional or axisymmetric, and three-dimensional turbulent internal flow problems. The method is based on an implicit upwinding relaxation scheme with an arbitrarily shaped conservative control volume. The compressible Reynolds-averaged Navier-Stokes equations are solved with a two-equation turbulence model. All these equations are expressed by using a non-orthogonal curvilinear co-ordinate system. The method is applied to study the compressible internal flow in modern power installations. It has been observed that predictions for two-dimensional and three-dimensional channels show very good agreement with experimental results.     

用混合网格及各向异性多重网格法求解三维可压紊流流动

采用混合网格求解紊流Navier Stokes方程。在物面附近采用柱状网格 ,其他区域则采用完全非结构网格。方程的求解采用Jamson的有限体积法 ,紊流模型采用两层Baldwin Lomax代数紊流模型。用各向异性多重网格法来加速解的收敛。数值算例表明 ,用混合网格及各向异性多重网格求解紊流流动是非常有效的