Anisotropic adaptive stabilized finite element solver for RANS models

Aerodynamic characteristics of various geometries are predicted using a finite element formulation coupled with several numerical techniques to ensure stability and accuracy of the method. First, an edge‐based error estimator and anisotropic mesh adaptation are used to detect automatically all flow features under the constraint of a fixed number of elements, thus controlling the computational cost. A variational multiscale‐stabilized finite element method is used to solve the incompressible Navier‐Stokes equations. Finally, the Spalart‐Allmaras turbulence model is solved using the streamline upwind Petrov‐Galerkin method. This paper is meant to show that the combination of anisotropic unsteady mesh adaptation with stabilized finite element methods provides an adequate framework for solving turbulent flows at high Reynolds numbers. The proposed method was validated on several test cases by confrontation with literature of both numerical and experimental results, in terms of accuracy on the prediction of the drag and lift coefficients as well as their evolution in time for unsteady cases.     

A high‐order compact finite‐difference lattice Boltzmann method for simulation of steady and unsteady incompressible flows

A high‐order compact finite‐difference lattice Boltzmann method (CFDLBM) is proposed and applied to accurately compute steady and unsteady incompressible flows. Herein, the spatial derivatives in the lattice Boltzmann equation are discretized by using the fourth‐order compact FD scheme, and the temporal term is discretized with the fourth‐order Runge–Kutta scheme to provide an accurate and efficient incompressible flow solver. A high‐order spectral‐type low‐pass compact filter is used to stabilize the numerical solution. An iterative initialization procedure is presented and applied to generate consistent initial conditions for the simulation of unsteady flows. A sensitivity study is also conducted to evaluate the effects of grid size, filtering, and procedure of boundary conditions implementation on accuracy and convergence rate of the solution. The accuracy and efficiency of the proposed solution procedure based on the CFDLBM method are also examined by comparison with the classical LBM for different flow conditions. Two test cases considered herein for validating the results of the incompressible steady flows are a two‐dimensional (2‐D) backward‐facing step and a 2‐D cavity at different Reynolds numbers. Results of these steady solutions computed by the CFDLBM are thoroughly compared with those of a compact FD Navier–Stokes flow solver. Three other test cases, namely, a 2‐D Couette flow, the Taylor's vortex problem, and the doubly periodic shear layers, are simulated to investigate the accuracy of the proposed scheme in solving unsteady incompressible flows. Results obtained for these test cases are in good agreement with the analytical solutions and also with the available numerical and experimental results. The study shows that the present solution methodology is robust, efficient, and accurate for solving steady and unsteady incompressible flow problems even at high Reynolds numbers. Copyright © 2014 John Wiley & Sons, Ltd.     

Application of a preconditioned high‐order accurate artificial compressibility‐based incompressible flow solver in wide range of Reynolds numbers

In the present study, the preconditioned incompressible Navier‐Stokes equations with the artificial compressibility method formulated in the generalized curvilinear coordinates are numerically solved by using a high‐order compact finite‐difference scheme for accurately and efficiently computing the incompressible flows in a wide range of Reynolds numbers. A fourth‐order compact finite‐difference scheme is utilized to accurately discretize the spatial derivative terms of the governing equations, and the time integration is carried out based on the dual time‐stepping method. The capability of the proposed solution methodology for the computations of the steady and unsteady incompressible viscous flows from very low to high Reynolds numbers is investigated through the simulation of different 2‐dimensional benchmark problems, and the results obtained are compared with the existing analytical, numerical, and experimental data. A sensitivity analysis is also performed to evaluate the effects of the size of the computational domain and other numerical parameters on the accuracy and performance of the solution algorithm. The present solution procedure is also extended to 3 dimensions and applied for computing the incompressible flow over a sphere. Indications are that the application of the preconditioning in the solution algorithm together with the high‐order discretization method in the generalized curvilinear coordinates provides an accurate and robust solution method for simulating the incompressible flows over practical geometries in a wide range of Reynolds numbers including the creeping flows.     

A finite element variational multiscale method for incompressible flows based on the construction of the projection basis functions

In this article, we present a finite element variational multiscale (VMS) method for incompressible flows based on the construction of projection basis functions and compare it with common VMS method, which is defined by a low‐order finite element space L_h on the same grid as X_h for the velocity deformation tensor and a stabilization parameter α. The best algorithmic feature of our method is to construct the projection basis functions at the element level with minimal additional cost to replace the global projection operator. Finally, we give some numerical simulations of the nonlinear flow problems to show good stability and accuracy properties of the method. Copyright © 2011 John Wiley & Sons, Ltd.     

An efficient parallel high-order compact scheme for the 3D incompressible Navier–Stokes equations

This article provides a strategy for solving incompressible turbulent flows, which combines compact finite difference schemes and parallel computing. The numerical features of this solver are the semi-implicit time advancement, the staggered arrangement of the variables and the fourth-order compact scheme discretisation. This is the usual way for solving accurately turbulent incompressible flows. We propose a new strategy for solving the Helmholtz/Poisson equations based on a parallel 2d-pencil decomposition of the diagonalisation method. The compact scheme derivatives are computed with the parallel diagonal dominant (PDD) algorithm, which achieves good parallel performances by introducing a bounded numerical error. We provide a new analysis of its effect on the numerical accuracy and conservation features. Several numerical experiments, including two simulations of turbulent flows, demonstrate that the PDD algorithm maintains the accuracy and conservation features, while conserving a good parallel performance, up to 4096 cores.     

基于动态混合网格的不可压非定常流计算方法

鱼类、昆虫等运动速度较低,对它们的数值模拟需要解决不可压问题.虚拟压缩方法通过在连续性方程中加入压强对虚拟时间的偏导数,从而把压力场和速度场耦合起来,解决了不可压缩流的计算问题.基于动态混合网格技术,利用双时间步方法耦合虚拟压缩方法来解决非定常不可压缩流的计算问题.为了加快每一虚拟时间步内的收敛速度,子迭代采用了高效的块LU-SGS方法,并且耦合了基于混合网格的多重网格方法.利用该方法数值模拟了不同雷诺数下的静止圆柱、振荡圆柱的绕流,得到了与实验和他人计算一致的结果.     

Introduction of turbulent model in a mixed finite volume/finite element method

The aim of this work is to present a new numerical method to compute turbulent flows in complex configurations. With this in view, a k-? model with wall functions has been introduced in a mixed finite volume/finite element method. The numerical method has been developed to deal with compressible flows but is also able to compute nearly incompressible flows. The physical model and the numerical method are first described, then validation results for an incompressible flow over a backward-facing step and for a supersonic flow over a compression ramp are presented. Comparisons are performed with experimental data and with other numerical results. These simulations show the ability of the present method to predict turbulent flows, and this method will be applied to simulate complex industrial flows (flow inside the combustion chamber of gas turbine engines). The main goal of this paper is not to test turbulence models, but to show that this numerical method is a solid base to introduce more sophisticated turbulence model.     

Implementing a high‐order accurate implicit operator scheme for solving steady incompressible viscous flows using artificial compressibility method

This paper uses a fourth‐order compact finite‐difference scheme for solving steady incompressible flows. The high‐order compact method applied is an alternating direction implicit operator scheme, which has been used by Ekaterinaris for computing two‐dimensional compressible flows. Herein, this numerical scheme is efficiently implemented to solve the incompressible Navier–Stokes equations in the primitive variables formulation using the artificial compressibility method. For space discretizing the convective fluxes, fourth‐order centered spatial accuracy of the implicit operators is efficiently obtained by performing compact space differentiation in which the method uses block‐tridiagonal matrix inversions. To stabilize the numerical solution, numerical dissipation terms and/or filters are used. In this study, the high‐order compact implicit operator scheme is also extended for computing three‐dimensional incompressible flows. The accuracy and efficiency of this high‐order compact method are demonstrated for different incompressible flow problems. A sensitivity study is also conducted to evaluate the effects of grid resolution and pseudocompressibility parameter on accuracy and convergence rate of the solution. The effects of filtering and numerical dissipation on the solution are also investigated. Test cases considered herein for validating the results are incompressible flows in a 2‐D backward facing step, a 2‐D cavity and a 3‐D cavity at different flow conditions. Results obtained for these cases are in good agreement with the available numerical and experimental results. The study shows that the scheme is robust, efficient and accurate for solving incompressible flow problems. Copyright © 2010 John Wiley & Sons, Ltd.     

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 finite volume unstructured multigrid method for efficient computation of unsteady incompressible viscous flows

An unstructured non‐nested multigrid method is presented for efficient simulation of unsteady incompressible Navier–Stokes flows. The Navier–Stokes solver is based on the artificial compressibility approach and a higher‐order characteristics‐based finite‐volume scheme on unstructured grids. Unsteady flow is calculated with an implicit dual time stepping scheme. For efficient computation of unsteady viscous flows over complex geometries, an unstructured multigrid method is developed to speed up the convergence rate of the dual time stepping calculation. The multigrid method is used to simulate the steady and unsteady incompressible viscous flows over a circular cylinder for validation and performance evaluation purposes. It is found that the multigrid method with three levels of grids results in a 75% reduction in CPU time for the steady flow calculation and 55% reduction for the unsteady flow calculation, compared with its single grid counterparts. The results obtained are compared with numerical solutions obtained by other researchers as well as experimental measurements wherever available and good agreements are obtained. Copyright © 2004 John Wiley & Sons, Ltd.     

Stabilized finite element solution to handle complex heat and fluid flows in industrial furnaces using the immersed volume method

We consider the numerical simulation of conjugate heat transfer, incompressible turbulent flows for multicomponents systems using a stabilized finite element method. We present an immersed volume approach for thermal coupling between fluids and solids for heating high‐alloy steel inside industrial furnaces. It consists in considering a single 3D grid of the furnace and solving one set of equations with different thermal properties. A distance function enables to define precisely the position and the interface of any objects inside the volume and to provide homogeneous physical and thermodynamic properties for each subdomain. An anisotropic mesh adaptation algorithm based on the variations of the distance function is then applied to ensure an accurate capture of the discontinuities that characterize the highly heterogeneous domain. The proposed method demonstrates the capability of the model to simulate an unsteady three‐dimensional heat transfers and turbulent flows in an industrial furnace with the presence of three conducting solid bodies. Copyright © 2010 John Wiley & Sons, Ltd.     

串列双圆柱绕流问题的数值模拟

本文运用有限体积方法,对绕串列放置的双圆柱的二维不可压缩流动进行了数值计算。为研究两圆柱不同间距对圆柱相互作用和尾流特征的影响,选取间距比Ｌ／Ｄ（Ｌ为两圆柱中心间的距离,Ｄ为圆柱直径）在１．５～５．０之间每隔０．５共八个有代表性的间距进行了计算模拟。计算均在Ｒｅ＝２００条件下进行。计算结果表明：对该绕流问题,流动特征在很大程度上取决于间距的大小。且间距存在一临界值,间距比从小于临界值变化到大于临界     

基于非结构网格的TTGC有限元格式的实现及在超声速流动中的应用

基于非结构网格系统,实现了时空三阶精度的TTGC有限元格式,并在三阶TTGC格式上发展了基于人工粘性的激波捕捉技术。在非结构网格下,采用这种方法对若干典型的超声速流动问题(SOD激波管、马赫数为3的前台阶流动以及马赫数为8的高超声速圆柱流动)进行了验证计算。结果表明,TTGC格式分辨率高,在粗糙网格下能够准确的模拟超声速流场中的激波、接触间断等复杂流动现象,并且能有效的控制间断附近的数值色散现象。与传统的有限体积方法相比,本文实现的TTGC有限元格式在模拟超声速流动问题方面具有格式精度高、数值耗散小等优点。     

A Taylor-Galerkin Method for Modeling Unsteady Fluid Flow Under its Own Weight

The free fluid-surface of incompressible creeping flows is analyzed using a finite element method. A pseudo-concentration (PC) function is introduced to determine the position of the free surface. The Taylor-Galerkin finite element method (TGFEM) is applied to solve the equation of the PC function. Nine-node quadratic interpolation is used for both PC and velocity. The unsteady flows of fluids moving of their own weight are analyzed using the proposed method.     

Study of a staggered fourth‐order compact scheme for unsteady incompressible viscous flows

The objective of this paper is the development and assessment of a fourth‐order compact scheme for unsteady incompressible viscous flows. A brief review of the main developments of compact and high‐order schemes for incompressible flows is given. A numerical method is then presented for the simulation of unsteady incompressible flows based on fourth‐order compact discretization with physical boundary conditions implemented directly into the scheme. The equations are discretized on a staggered Cartesian non‐uniform grid and preserve a form of kinetic energy in the inviscid limit when a skew‐symmetric form of the convective terms is used. The accuracy and efficiency of the method are demonstrated in several inviscid and viscous flow problems. Results obtained with different combinations of second‐ and fourth‐order spatial discretizations and together with either the skew‐symmetric or divergence form of the convective term are compared. The performance of these schemes is further demonstrated by two challenging flow problems, linear instability in plane channel flow and a two‐dimensional dipole–wall interaction. Results show that the compact scheme is efficient and that the divergence and skew‐symmetric forms of the convective terms produce very similar results. In some but not all cases, a gain in accuracy and computational time is obtained with a high‐order discretization of only the convective and diffusive terms. Finally, the benefits of compact schemes with respect to second‐order schemes is discussed in the case of the fully developed turbulent channel flow. Copyright © 2008 John Wiley & Sons, Ltd.     

A mixed‐interpolation finite element method for incompressible thermal flows of electrically conducting fluids

A new mixed‐interpolation finite element method is presented for the two‐dimensional numerical simulation of incompressible magnetohydrodynamic (MHD) flows which involve convective heat transfer. The proposed method applies the nodal shape functions, which are locally defined in nine‐node elements, for the discretization of the Navier–Stokes and energy equations, and the vector shape functions, which are locally defined in four‐node elements, for the discretization of the electromagnetic field equations. The use of the vector shape functions allows the solenoidal condition on the magnetic field to be automatically satisfied in each four‐node element. In addition, efficient approximation procedures for the calculation of the integrals in the discretized equations are adopted to achieve high‐speed computation. With the use of the proposed numerical scheme, MHD channel flow and MHD natural convection under a constant applied magnetic field are simulated at different Hartmann numbers. The accuracy and robustness of the method are verified through these numerical tests in which both undistorted and distorted meshes are employed for comparison of numerical solutions. Furthermore, it is shown that the calculation speed for the proposed scheme is much higher compared with that for a conventional numerical integration scheme under the condition of almost the same memory consumption. Copyright © 2016 John Wiley & Sons, Ltd.     

A mass-matrix formulation of unsteady fluctuation splitting schemes consistent with Roe’s parameter vector

A mass-matrix formulation of the fluctuation splitting schemes for solving compressible, unsteady flows is proposed. This formulation is consistent with the conservative linearisation based on parameter vector and allows to extend to unsteady flows the ‘invariance under similarity transformations’ property that had been shown to hold for the steady version of the schemes. Second-order time accuracy is achieved using a Petrov–Galerkin finite element interpretation of the fluctuation splitting schemes. The approach may however be readily applicable to all other time-accurate fluctuation splitting formulations that have been so far proposed in the literature. Applications of the proposed methodology to two- and three-dimensional, inviscid and viscous compressible flows are reported and discussed in the paper.     

Finite element analysis of incompressible laminar boundary layer flows

A numerical procedure was developed to solve the two-dimensional and axisymmetric incompressible laminar boundary layer equations using the semi-discrete Galerkin finite element method. Linear Lagrangian, quadratic Lagrangian, and cubic Hermite interpolating polynomials were used for the finite element discretization; the first-order, the second-order backward difference approximation, and the Crank-Nicolson method were used for the system of non-linear ordinary differential equations; the Picard iteration and the Newton-Raphson technique were used to solve the resulting non-linear algebraic system of equations. Conservation of mass is treated as a constraint condition in the procedure; hence, it is integrated numerically along the solution line while marching along the time-like co-ordinate. Among the numerical schemes tested, the Picard iteration technique used with the quadratic Lagrangian polynomials and the second-order backward difference approximation case turned out to be the most efficient to achieve the same accuracy. The advantages of the method developed lie in its coarse grid accuracy, global computational efficiency, and wide applicability to most situations that may arise in incompressible laminar boundary layer flows.     

Finite element approximation for single-phase multicomponent flows

With the objective of performing computational simulations of mixture problems, this work employed a mechanical modeling of single-phase incompressible multicomponent flows able to deal with geometric and material non-linearities. This model is based on conservative laws of mass and momentum in continuum mechanics, assuming the mixture as a superposition of a number of continuous media, each one with a variable volume fraction field. Using appropriate hypothesis, the model was formulated as a set of non-linear partial differential equations and associated boundary conditions. This set was approximated by a stabilized finite element method, based on a Galerkin/least-squares scheme, in order to circumvent Babu ka–Brezzi condition and to generate stable approximations even in highly advective situations. Some preliminary numerical results have been obtained for non-Newtonian axial injections in backward facing step incompressible flows.     

A comparative study of GLS finite elements with velocity and pressure equally interpolated for solving incompressible viscous flows

A comparative study of the bi‐linear and bi‐quadratic quadrilateral elements and the quadratic triangular element for solving incompressible viscous flows is presented. These elements make use of the stabilized finite element formulation of the Galerkin/least‐squares method to simulate the flows, with the pressure and velocity fields interpolated with equal orders. The tangent matrices are explicitly derived and the Newton–Raphson algorithm is employed to solve the resulting nonlinear equations. The numerical solutions of the classical lid‐driven cavity flow problem are obtained for Reynolds numbers between 1000 and 20 000 and the accuracy and converging rate of the different elements are compared. The influence on the numerical solution of the least square of incompressible condition is also studied. The numerical example shows that the quadratic triangular element exhibits a better compromise between accuracy and converging rate than the other two elements. Copyright © 2008 John Wiley & Sons, Ltd.