Time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process for incompressible steady flow

The time‐related element‐free Taylor–Galerkin method with non‐splitting decoupling process (EFTG‐NSD) is proposed for the simulation of steady flows. The goal of the present paper is twofold. One is to raise the efficiency of the time‐related methods for solving steady flow problems, and the other is to obtain a good stability. The EFTG‐NSD method, which uses the time‐related Navier–Stokes equations to describe steady flows, does not care about the intermediate process and obtains solution of steady flows through time marching. Different from the classical time‐related fractional step methods, the EFTG‐NSD method decouples the Navier–Stokes equations without any operator‐splitting and correction. Because the elimination of correction at each iteration step reduces the computation cost, the EFTG‐NSD method possesses higher computation efficiency. In addition, the EFTG‐NSD method has a good stability due to the use of the Taylor–Galerkin formula in time and space discretization. Furthermore, the method combining element‐free Galerkin method with Taylor–Galerkin method is an important supplement of the element‐free Galerkin method for solving flow problems. Copyright © 2011 John Wiley & Sons, Ltd.     

A fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes

Simulation of 1D steady flow covers a wide range of practical applications, such as rivers, pipes and hydraulic structures. Various flow patterns coexist in such situations: free surface flows (supercritical, subcritical and hydraulic jump), pressurized flows as well as mixed flows. As a result, development of a unified 1D model for all the situations of interest in civil engineering remains challenging. In this paper, a fast universal solver for 1D continuous and discontinuous steady flows in rivers and pipes is set up and assessed. Developments are initiated from an original unified mathematical model using the Saint‐Venant equations. Application of these equations, originally dedicated to free‐surface flow, is extended to pressurized flow by means of the Preissmann slot model. In particular, an original negative slot is developed in order to handle sub‐atmospheric pressurized flow. Next, the full unsteady model is simplified under the assumption of steadiness and reformulated into a single pseudo‐unsteady differential equation. The derived pseudo‐unsteady formulation aims at keeping the hyperbolic feature of the equation. Stability analysis of the differential equation suggests a unique splitting for the finite volume scheme whatever the flow conditions. The numerical scheme obtained is a universal Flux Vector Splitting which shows robustness and simplicity. Accuracy and performance of the new methodology is assessed by comparison with analytical and experimental results. Copyright © 2009 John Wiley & Sons, Ltd.     

Two‐dimensional implicit time‐dependent calculations on adaptive unstructured meshes with time evolving boundaries

An implicit multigrid‐driven algorithm for two‐dimensional incompressible laminar viscous flows has been coupled with a solution adaptation method and a mesh movement method for boundary movement. Time‐dependent calculations are performed implicitly by regarding each time step as a steady‐state problem in pseudo‐time. The method of artificial compressibility is used to solve the flow equations. The solution mesh adaptation method performs local mesh refinement using an incremental Delaunay algorithm and mesh coarsening by means of edge collapse. Mesh movement is achieved by modeling the computational domain as an elastic solid and solving the equilibrium equations for the stress field. The solution adaptation method has been validated by comparison with experimental results and other computational results for low Reynolds number flow over a shedding circular cylinder. Preliminary validation of the mesh movement method has been demonstrated by a comparison with experimental results of an oscillating airfoil and with computational results for an oscillating cylinder. Copyright © 2005 John Wiley & Sons, Ltd.     

A domain decomposition approach to finite volume solutions of the Euler equations on unstructured triangular meshes

We report on our recent efforts on the formulation and the evaluation of a domain decomposition algorithm for the parallel solution of two‐dimensional compressible inviscid flows. The starting point is a flow solver for the Euler equations, which is based on a mixed finite element/finite volume formulation on unstructured triangular meshes. Time integration of the resulting semi‐discrete equations is obtained using a linearized backward Euler implicit scheme. As a result, each pseudo‐time step requires the solution of a sparse linear system for the flow variables. In this study, a non‐overlapping domain decomposition algorithm is used for advancing the solution at each implicit time step. First, we formulate an additive Schwarz algorithm using appropriate matching conditions at the subdomain interfaces. In accordance with the hyperbolic nature of the Euler equations, these transmission conditions are Dirichlet conditions for the characteristic variables corresponding to incoming waves. Then, we introduce interface operators that allow us to express the domain decomposition algorithm as a Richardson‐type iteration on the interface unknowns. Algebraically speaking, the Schwarz algorithm is equivalent to a Jacobi iteration applied to 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 our approach, the interface unknowns are numerical (normal) fluxes. Copyright © 2001 John Wiley & Sons, Ltd.     

A promising boundary element formulation for three‐dimensional viscous flow

In this paper, a new set of boundary‐domain integral equations is derived from the continuity and momentum equations for three‐dimensional viscous flows. The primary variables involved in these integral equations are velocity, traction, and pressure. The final system of equations entering the iteration procedure only involves velocities and tractions as unknowns. In the use of the continuity equation, a complex‐variable technique is used to compute the divergence of velocity for internal points, while the traction‐recovery method is adopted for boundary points. Although the derived equations are valid for steady, unsteady, compressible, and incompressible problems, the numerical implementation is only focused on steady incompressible flows. Two commonly cited numerical examples and one practical pipe flow problem are presented to validate the derived equations. Copyright © 2004 John Wiley & Sons, Ltd.     

Incompressible flows with combustion simulated by a preconditioning method using multigrid acceleration and MUSCL reconstruction

A numerical algorithm for the steady state solution of three‐dimensional incompressible flows is presented. A preconditioned time marching scheme is applied to the conservative form of the governing equations. The preconditioning matrix multiplies the time derivatives of the system and circumvents the eigenvalue‐caused stiffness at low speed. The formulation is suitable for constant density flows and for flows where the density depends on non‐passive scalars, such as in low‐speed combustion applications. The k–ε model accounts for turbulent transport effects. A cell‐centred finite volume formulation with a Runge–Kutta time stepping scheme for the primitive variables is used. Second‐order spatial accuracy is achieved by developing for the preconditioned system an approximate Riemann solver with MUSCL reconstruction. A multi‐grid technique coupled with local time stepping and implicit residual smoothing is used to accelerate the convergence to the steady state solution. The convergence behaviour and the validation of the predicted solutions are examined for laminar and turbulent constant density flows and for a turbulent non‐premixed flame simulated by a presumed probability density function (PDF) model. Copyright © 2001 John Wiley & Sons, Ltd.     

Implicit preconditioned high‐order compact scheme for the simulation of the three‐dimensional incompressible Navier–Stokes equations with pseudo‐compressibility method

This paper combines the pseudo‐compressibility procedure, the preconditioning technique for accelerating the time marching for stiff hyperbolic equations, and high‐order accurate central compact scheme to establish the code for efficiently and accurately solving incompressible flows numerically based on the finite difference discretization. The spatial scheme consists of the sixth‐order compact scheme and 10th‐order numerical filter operator for guaranteeing computational stability. The preconditioned pseudo‐compressible Navier–Stokes equations are marched temporally using the implicit lower–upper symmetric Gauss–Seidel time integration method, and the time accuracy is improved by the dual‐time step method for the unsteady problems. The efficiency and reliability of the present procedure are demonstrated by applications to Taylor decaying vortices phenomena, double periodic shear layer rolling‐up problem, laminar flow over a flat plate, low Reynolds number unsteady flow around a circular cylinder at Re = 200, high Reynolds number turbulence flow past the S809 airfoil, and the three‐dimensional flows through two 90°curved ducts of square and circular cross sections, respectively. It is found that the numerical results of the present algorithm are in good agreement with theoretical solutions or experimental data. Copyright © 2011 John Wiley & Sons, Ltd.     

On higher‐order mixed FEM for low Mach number flows: application to a natural convection benchmark problem

We consider higher‐order mixed finite elements with continuous pressures for the computation of stationary compressible flows at low Mach number. The proposed approach is based on a fully coupled treatment of the governing equations and therefore, for steady‐state calculations, does not rely on time‐stepping techniques. The non‐linear problem is solved by means of a quasi‐Newton iteration. The strongly coupled system resulting from higher‐order discretization of the linearized equations requires adequate solvers. We propose a new scheme based on multigrid methods with varying FEM ansatz orders on the grid hierarchy as well as multiplicative smoothers based on blocking techniques. Computational results are described for a benchmark configuration including a flow with heat transfer in the low Mach number regime. Furthermore, the issue of anisotropic grids is addressed in that context. Copyright © 2003 John Wiley & Sons, Ltd.     

Gauge finite element method for incompressible flows

A finite element method for computing viscous incompressible flows based on the gauge formulation introduced in [Weinan E, Liu J‐G. Gauge method for viscous incompressible flows. Journal of Computational Physics (submitted)] is presented. This formulation replaces the pressure by a gauge variable. This new gauge variable is a numerical tool and differs from the standard gauge variable that arises from decomposing a compressible velocity field. It has the advantage that an additional boundary condition can be assigned to the gauge variable, thus eliminating the issue of a pressure boundary condition associated with the original primitive variable formulation. The computational task is then reduced to solving standard heat and Poisson equations, which are approximated by straightforward, piecewise linear (or higher‐order) finite elements. This method can achieve high‐order accuracy at a cost comparable with that of solving standard heat and Poisson equations. It is naturally adapted to complex geometry and it is much simpler than traditional finite element methods for incompressible flows. Several numerical examples on both structured and unstructured grids are presented. Copyright © 2000 John Wiley & Sons, Ltd.     

Comparison of spectral and finite element methods applied to the study of the core‐annular flow in an undulating tube

A Galerkin/finite element and a pseudo‐spectral method, in conjunction with the primitive (velocity‐pressure) and streamfunction‐vorticity formulations, are tested for solving the two‐phase flow in a tube, which has a periodically varying, circular cross section. Two immiscible, incompressible, Newtonian fluids are arranged so that one of them is around the axis of the tube (core fluid) and the other one surrounds it (annular fluid). The physical and flow parameters are such that the interface between the two fluids remains continuous and single‐valued. This arrangement is usually referred to as Core‐Annular flow. A non‐orthogonal mapping is used to transform the uneven tube shape and the unknown, time dependent interface to fixed, cylindrical surfaces. With both methods and formulations, steady states are calculated first using the Newton–Raphson method. The most dangerous eigenvalues of the related linear stability problem are calculated using the Arnoldi method, and dynamic simulations are carried out using the implicit Euler method. It is shown that with a smooth tube shape the pseudo‐spectral method exhibits exponential convergence, whereas the finite element method exhibits algebraic convergence, albeit of higher order than expected from the relevant theory. Thus the former method, especially when coupled with the streamfunction‐vorticity formulation, is much more efficient. The finite element method becomes more advantageous when the tube shape contains a cusp, in which case the convergence rate of the pseudo‐spectral method deteriorates exhibiting algebraic convergence with the number of the axial spectral modes, whereas the convergence rate of the finite element method remains unaffected. Copyright © 2002 John Wiley & Sons, Ltd.     

Velocity–pressure coupling in finite difference formulations for the Navier–Stokes equations

A new numerical procedure for solving the two‐dimensional, steady, incompressible, viscous flow equations on a staggered Cartesian grid is presented in this paper. The proposed methodology is finite difference based, but essentially takes advantage of the best features of two well‐established numerical formulations, the finite difference and finite volume methods. Some weaknesses of the finite difference approach are removed by exploiting the strengths of the finite volume method. In particular, the issue of velocity–pressure coupling is dealt with in the proposed finite difference formulation by developing a pressure correction equation using the SIMPLE approach commonly used in finite volume formulations. However, since this is purely a finite difference formulation, numerical approximation of fluxes is not required. Results presented in this paper are based on first‐ and second‐order upwind schemes for the convective terms. This new formulation is validated against experimental and other numerical data for well‐known benchmark problems, namely developing laminar flow in a straight duct, flow over a backward‐facing step, and lid‐driven cavity flow. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

One‐shot pseudo‐time method for aerodynamic shape optimization using the Navier–Stokes equations

This paper presents a numerical method for aerodynamic shape optimization problems in compressible viscous flow. It is based on simultaneous pseudo‐time stepping in which stationary states are obtained by solving the pseudo‐stationary system of equations representing the state, costate and design equations. The main advantages of this method are that it blends in nicely with previously existing pseudo‐time‐stepping methods for the state and the costate equations, that it requires no additional globalization in the design space, and that a preconditioner can be used for convergence acceleration which stems from the reduced SQP methods. For design examples of 2D problems, the overall cost of computation can be reduced to less than 2 times the forward simulation runs. Copyright © 2011 John Wiley & Sons, Ltd.     

Implicit weighted essentially non‐oscillatory schemes for the incompressible Navier–Stokes equations

A class of lower–upper/approximate factorization (LUAF) implicit weighted essentially non‐oscillatory (ENO; WENO) schemes for solving the two‐dimensional incompressible Navier–Stokes equations in a generalized co‐ordinate system is presented. The algorithm is based on the artificial compressibility formulation, and symmetric Gauss–Seidel relaxation is used for computing steady state solutions while symmetric successive overrelaxation is used for treating time‐dependent flows. WENO spatial operators are employed for inviscid fluxes and central differencing for viscous fluxes. Internal and external viscous flow test problems are presented to verify the numerical schemes. The use of a WENO spatial operator not only enhances the accuracy of solutions but also improves the convergence rate for the steady state computation as compared with using the ENO counterpart. It is found that the present solutions compare well with exact solutions, experimental data and other numerical results. Copyright © 1999 John Wiley & Sons, Ltd.     

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.     

A study of gridless method with dynamic clouds of points for solving unsteady CFD problems in aerodynamics

When solving unsteady computational fluid dynamics problems in aerodynamics with a gridless method, a cloud of points is usually required to be regenerated due to its accommodation to moving boundaries. In order to handle this problem conveniently, a fast dynamic cloud method based on Delaunay graph mapping strategy is proposed in this paper. A dynamic cloud method makes use of algebraic mapping principles and therefore points can be accurately redistributed in the flow field without any iteration. In this way, the structure of the gridless clouds is not necessarily changed so that the clouds regeneration can be avoided successfully. The spatial derivatives of the mathematical modeling of the flow are directly determined by using weighted least‐squares method in each cloud of points, and then numerical fluxes can be obtained. A dual time‐stepping method is further implemented to advance the two‐dimensional Euler equations in arbitrary Lagarangian–Eulerian formulation in time. Finally, unsteady transonic flows over two different oscillating airfoils are simulated with the above method and results obtained are in good agreement with the experimental data. Copyright © 2009 John Wiley & Sons, Ltd.     

A momentum coupling method for the unsteady incompressible Navier–Stokes equations on the staggered grid

A new numerical method is developed to efficiently solve the unsteady incompressible Navier–Stokes equations with second-order accuracy in time and space. In contrast to the SIMPLE algorithms, the present formulation directly solves the discrete x- and y-momentum equations in a coupled form. It is found that the present implicit formulation retrieves some cross convection terms overlooked by the conventional iterative methods, which contribute to accuracy and fast convergence. The finite volume method is applied on the fully staggered grid to solve the vector-form momentum equations. The preconditioned conjugate gradient squared method (PCGS) has proved very efficient in solving the associate linearized large, sparse block-matrix system. Comparison with the SIMPLE algorithm has indicated that the present momentum coupling method is fast and robust in solving unsteady as well as steady viscous flow problems. © 1998 John Wiley & Sons, Ltd.     

A high‐order finite difference method for incompressible fluid turbulence simulations

A Hermitian–Fourier numerical method for solving the Navier–Stokes equations with one non‐homogeneous direction had been presented by Schiestel and Viazzo (Internat. J. Comput. Fluids 1995; 24 (6):739). In the present paper, an extension of the method is devised for solving problems with two non‐homogeneous directions. This extension is indeed not trivial since new algorithms will be necessary, in particular for pressure calculation. The method uses Hermitian finite differences in the non‐periodic directions whereas Fourier pseudo‐spectral developments are used in the remaining periodic direction. Pressure–velocity coupling is solved by a simplified Poisson equation for the pressure correction using direct method of solution that preserves Hermitian accuracy for pressure. The turbulent flow after a backward facing step has been used as a test case to show the capabilities of the method. The applications in view are mainly concerning the numerical simulation of turbulent and transitional flows. Copyright © 2003 John Wiley & Sons, Ltd.     

预处理方法在全速度流场中的应用研究

应用预处理方法求解二维可压缩Navier-Stokes方程,发展出一套既适用于低速流动,也可用于可压缩流动的全速度流场解算方法.空间格式选用Ausm类格式,并采用多步Runge-Kutta时间推进方法.数值模拟了鼓包(Bump),RAE2822翼型,多段高升力翼型的不同速度绕流流场,并与实验数据进行了对比.结果表明预处...     

Numerical methods for conjunctive two‐dimensional surface and three‐dimensional sub‐surface flows

Sophisticated catchment runoff problems necessitate conjunctive modeling of overland flow and sub‐surface flow. In this paper, finite difference numerical methods are studied for simulation of catchment runoff of two‐dimensional surface flow interacting with three‐dimensional unsaturated and saturated sub‐surface flows. The equations representing the flows are mathematically classified as a type of heat diffusion equation. Therefore, two‐ and three‐dimensional numerical methods for heat diffusion equations were investigated for applications to the surface and sub‐surface flow sub‐models in terms of accuracy, stability, and calculation time. The methods are the purely explicit method, Saul'yev's methods, the alternating direction explicit (ADE) methods, and the alternating direction implicit (ADI) methods. The methods are first examined on surface and sub‐surface flows separately; subsequently, 12 selected combinations of methods were investigated for modeling the conjunctive flows. Saul'yev's downstream (S‐d) method was found to be the preferred method for two‐dimensional surface flow modeling, whereas the ADE method of Barakat and Clark is a less accurate, stable alternative. For the three‐dimensional sub‐surface flow model, the ADE method of Larkin (ADE‐L) and Brian's ADI method are unconditionally stable and more accurate than the other methods. The calculations of the conjunctive models utilizing the S‐d surface flow sub‐model give excellent results and confirm the expectation that the errors of the surface and sub‐surface sub‐models interact. The surface sub‐model dominates the accuracy and stability of the conjunctive model, whereas the sub‐surface sub‐model dominates the calculation time, suggesting the desirability of using a smaller time increment for the surface sub‐model. Copyright © 2000 John Wiley & Sons, Ltd.