On the implementation of symmetric and antisymmetric periodic boundary conditions for incompressible flow

In this paper we consider symmetric and antisymmetric periodic boundary conditions for flows governed by the incompressible Navier-Stokes equations. Classical periodic boundary conditions are studied as well as symmetric and antisymmetric periodic boundary conditions in which there is a pressure difference between inlet and outlet. The implementation of this type of boundary conditions in a finite element code using the penalty function formulation is treated and also the implementation in a finite volume code based on pressure correction. The methods are demonstrated by computation of a flow through a staggered tube bundle.     

Numerical solution of the unsteady Euler equations for airfoils using approximate boundary conditions

This paper presents an efficient numerical method for solving the unsteady Euler equations on stationary rectilinear grids. Boundary conditions on the surface of an airfoil are implemented by using their first-order expansions on the mean chord line. The method is not restricted to flows with small disturbances since there are no restrictions on the mean angle of attack of the airfoil. The mathematical formulation and the numerical implementation of the wall boundary conditions in a fully implicit time-accurate finite-volume Euler scheme are described. Unsteady transonic flows about an oscillating NACA 0012 airfoil are calculated. Computational results compare well with Euler solutions by the full boundary conditions on a body-fitted curvilinear grid and published experimental data. This study establishes the feasibility for computing unsteady fluid-structure interaction problems, where the use of a stationary rectilinear grid offers substantial advantages in saving computer time and program design since it does not require the generation and implementation of time-dependent body-fitted grids.     

Approximate nonreflecting inflow-outflow boundary conditions for the calculation of navier-stokes equations in internal flows

In this paper, the nonreflecting boundary conditions based upon fundamental ideas of the linear analysis are developed for gas dynamic equations, and the modified boundary conditions for Navier-Stokes equations are proposed as a substitute of the nonreflecting boundary conditions inside boundary layers near rigid walls. These derived boundary conditions are then applied to calculations both for the Euler equations and the Navier-Stokes equations to determine if they can produce acceptable results for the subsonic flows in channels. The numerical results obtained by an implicit second-order upwind difference scheme show the effectiveness and generality of the boundary conditions. Furthermore, the formulae and the analysis performed here may be extended to three dimensional problems. recommended by Prof. Cui Erjie     

Solution of three-dimensional viscous incompressible flows by a multi-grid method

The three-dimensional Navier-Stokes equations for viscous incompressible fluids are discretized on staggered or non-staggered grids. The system of finite-difference equations is solved by a multi-grid method. The method and some possible sources of difficulties and their remedies are described. The numerical algorithm has been applied to the computations of flows in ducts for a range of Reynolds numbers up to 2000. As outflow boundary conditions, either the fully developed flow profile (Dirichlet condition) or parabolic conditions have been applied. The multi-grid method has a fast rate of convergence (with both types of boundary conditions), and it is not sensitive to the number of mesh points and the Reynolds number. The numerical solution, using parabolic boundary conditions, is insensitive to the location of the outflow boundary, even for large Reynolds numbers, in contrast to the solution with Dirichlet boundary conditions.     

A mixture differential quadrature method for solving two-dimensional incompressible Navier-Stokes equations

Ｔｈｅｄｉｆｆｅｒｅｎｔｉａｌｑｕａｄｒａｔｕｒｅｍｅｔｈｏｄ（ＤＱＭ）ｐｒｏｐｏｓｅｄｂｙＲ．Ｂｅｌｌｍａｎ［１,２］ｈａｓｂｅｅｎｓｕｃｃｅｓｓｆｕｌｌｙｅｍｐｌｏｙｅｄｉｎｎｕｍｅｒｉｃａｌｃｏｍｐｕｔａｔｉｏｎｓｏｆｐｒｏｂｌｅｍｓｉｎｅｎｇｉｎｅｅｒｉｎｇａｎｄｐｈｙｓｉｃａｌｓｃｉｅｎｃｅ．ＢｅｃａｕｓｅｔｈｅｉｎｆｏｒｍａｔｉｏｎｏｎａｌｌｇｒｉｄｐｏｉｎｔｓｉｓｕｓｅｄｔｏｆｉｔｔｈｅｄｅｒｉｖａｔｉｖｅｓａｔｇｒｉｄｐｏｉｎｔｓｉｎｔｈｅＤＱＭ,ｉｔｉｓｅｎｏｕｇｈｔｏｏｂｔａ…     

A new method for computing solenoidal vector fields on arbitrary regions

We develop a new method for the efficient calculation of solenoidal vector fields on general regions. The method takes advantage of fast direct methods and uses boundary integral equations to satisfy boundary conditions. For the latter we give an effective scheme for computing far-field boundary influences (based on discrete charges). Examples and numerical results are given. The method is applicable to incompressible Navier-Stokes calculations.     

Vortex Systems in a Viscous Fluid in the Neighborhood of a Corner Point on the Boundary

The expansion of the solution of the Navier-Stokes equations for steady-state plane-parallel incompressible fluid flows in a series in powers of the Reynolds number and the subjection of this series to no-slip conditions on the rectilinear boundaries in the neighborhood of their intersection point leads to the asymptotic form of the solution in the neighborhood of that point. The use of the leading part of the asymptotic form obtained as the boundary condition at a certain distance from the corner point makes it possible to formulate boundary value problems for the Navier-Stokes equations in closed regions. Examples of the numerical solution of such problems illustrate the formation of infinite vortex systems in the neighborhood of the corner point on the boundary of the flow region.     

Development of an Artificial Compressibility Methodology with Implicit LU-SGS Method

A numerical method has been developed to solve the steady and unsteady incompressible Navier-Stokes equations in a two-dimensional, curvilinear coordinate system. The solution procedure is based on the method of artificial compressibility and uses a third-order flux-difference splitting upwind differencing scheme for convective terms and second-order center difference for viscous terms. A time-accurate scheme for unsteady incompressible flows is achieved by using an implicit real time discretization and a dual-time approach, which introduces pseudo-unsteady terms into both the mass conservation equation and momentum equations. An efficient fully implicit algorithm LU-SGS, which was originally derived for the compressible Eulur and Navier-Stokes equations by Jameson and Toon [1], is developed for the pseudo-compressibility formulation of the two dimensional incompressible Navier-Stokes equations for both steady and unsteady flows. A variety of computed results are presented to validate the present scheme. Numerical solutions for steady flow in a square lid-driven cavity and over a backward facing step and for unsteady flow in a square driven cavity with an oscillating lid and in a circular tube with a smooth expansion are respectively presented and compared with experimental data or other numerical results.     

A matrix-free high-order discontinuous Galerkin compressible Navier-Stokes solver: A performance comparison of compressible and incompressible formulations for turbulent incompressible flows

Both compressible and incompressible Navier-Stokes solvers can be used and are used to solve incompressible turbulent flow problems. In the compressible case, the Mach number is then considered as a solver parameter that is set to a small value, M ≈0.1, in order to mimic incompressible flows. This strategy is widely used for high-order discontinuous Galerkin (DG) discretizations of the compressible Navier-Stokes equations. The present work raises the question regarding the computational efficiency of compressible DG solvers as compared to an incompressible formulation. Our contributions to the state of the art are twofold: Firstly, we present a high-performance DG solver for the compressible Navier-Stokes equations based on a highly efficient matrix-free implementation that targets modern cache-based multicore architectures with Flop/Byte ratios significantly larger than 1. The performance results presented in this work focus on the node-level performance, and our results suggest that there is great potential for further performance improvements for current state-of-the-art DG implementations of the compressible Navier-Stokes equations. Secondly, this compressible Navier-Stokes solver is put into perspective by comparing it to an incompressible DG solver that uses the same matrix-free implementation. We discuss algorithmic differences between both solution strategies and present an in-depth numerical investigation of the performance. The considered benchmark test cases are the three-dimensional Taylor-Green vortex problem as a representative of transitional flows and the turbulent channel flow problem as a representative of wall-bounded turbulent flows. The results indicate a clear performance advantage of the incompressible formulation over the compressible one.     

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。     

Investigation of solution of Navier–Stokes equations using a variational formulation

A variational formulation for the solution of two dimensional, incompressible viscous flows has been developed by one of the authors.¹ The main objective of the present paper is to demonstrate the applicability of this approach for the solution of practical problems and in particular to investigate the introduction of boundary conditions to the Navier-Stokes equations through a variational formulation. The application of boundary conditions for typical internal and external flow problems is presented. Sample cases include flow around a cylinder and flow through a stepped channel. Quadrilateral, bilinear isoparametric elements are utilized in the formulation. A single-step, implicit, and fully coupled numerical integration scheme based on the variational principle is employed. Presented results include sample cases with different Reynolds numbers for laminar and turbulent flows. Turbulence is modelled using a simple mixing length model. Numerical results show good agreement with existing solutions.     

A high-order nodal discontinuous Galerkin method to solve preconditioned multiphase Euler/Navier-Stokes equations for inviscid/viscous cavitating flows

In this study, a high-order accurate numerical method is applied and examined for the simulation of the inviscid/viscous cavitating flows by solving the preconditioned multiphase Euler/Navier-Stokes equations on triangle elements. The formulation used here is based on the homogeneous equilibrium model considering the continuity and momentum equations together with the transport equation for the vapor phase with applying appropriate mass transfer terms for calculating the evaporation/condensation of the liquid/vapor phase. The spatial derivative terms in the resulting system of equations are discretized by the nodal discontinuous Galerkin method (NDGM) and an implicit dual-time stepping method is used for the time integration. An artificial viscosity approach is implemented and assessed for capturing the steep discontinuities in the interface between the two phases. The accuracy and robustness of the proposed method in solving the preconditioned multiphase Euler/Navier-Stokes equations are examined by the simulation of different two-dimensional and axisymmetric cavitating flows. A sensitivity study is also performed to examine the effects of different numerical parameters on the accuracy and performance of the solution of the NDGM. Indications are that the solution methodology proposed and applied here is based on the NDGM with the implicit dual-time stepping method and the artificial viscosity approach is accurate and robust for the simulation of the inviscid and viscous cavitating flows.     

Combined immersed boundary method and multiple-relaxation-time lattice Boltzmann flux solver for numerical simulations of incompressible flows

A method combining the immersed boundary technique and a multirelaxation-time(MRT) lattice Boltzmann flux solver(LBFS) is presented for numerical simulation of incompressible flows over circular and elliptic cylinders and NACA 0012 Airfoil. The method uses a simple Cartesian mesh to simulate flows past immersed complicated bodies. With the Chapman-Enskog expansion analysis, a transform is performed between the Navier-Stokes and lattice Boltzmann equations(LBEs). The LBFS is used to discretize the macroscopic differential equations with a finite volume method and evaluate the interface fluxes through local reconstruction of the lattice Boltzmann solution.The immersed boundary technique is used to correct the intermediate velocity around the solid boundary to satisfy the no-slip boundary condition. Agreement of simulation results with the data found in the literature shows reliability of the proposed method in simulating laminar flows on a Cartesian mesh.     

Development of an artificial compressibility methodology using flux vector splitting

An implicit, upwind arithmetic scheme that is efficient for the solution of laminar, steady, incompressible, two-dimensional flow fields in a generalised co-ordinate system is presented in this paper. The developed algorithm is based on the extended flux-vector-splitting (FVS) method for solving incompressible flow fields. As in the case of compressible flows, the FVS method consists of the decomposition of the convective fluxes into positive and negative parts that transmit information from the upstream and downstream flow field respectively. The extension of this method to the solution of incompressible flows is achieved by the method of artificial compressibility, whereby an artificial time derivative of the pressure is added to the continuity equation. In this way the incompressible equations take on a hyperbolic character with pseudopressure waves propagating with finite speed. In such problems the ‘information’ inside the field is transmitted along its characteristic curves. In this sense, we can use upwind schemes to represent the finite volume scheme of the problem's governing equations. For the representation of the problem variables at the cell faces, upwind schemes up to third order of accuracy are used, while for the development of a time-iterative procedure a first-order-accurate Euler backward-time difference scheme is used and a second-order central differencing for the shear stresses is presented. The discretized Navier–Stokes equations are solved by an implicit unfactored method using Newton iterations and Gauss–Siedel relaxation. To validate the derived arithmetical results against experimental data and other numerical solutions, various laminar flows with known behaviour from the literature are examined. © 1997 John Wiley & Sons, Ltd.     

Time‐accurate intermediate boundary conditions for large eddy simulations based on projection methods

Projection methods are among the most adopted procedures for solving the Navier–Stokes equations system for incompressible flows. In order to simplify the numerical procedures, the pressure–velocity de‐coupling is often obtained by adopting a fractional time‐step method. In a specific formulation, suitable for the incompressible flows equations, it is based on a formal decomposition of the momentum equation, which is related to the Helmholtz–Hodge Decomposition theorem of a vector field in a finite domain. Owing to the continuity constraint also in large eddy simulation of turbulence, as happens for laminar solutions, the filtered pressure characterizes itself only as a Lagrange multiplier, not a thermodynamic state variable. The paper illustrates the implications of adopting such procedures when the decoupling is performed onto the filtered equations system. This task is particularly complicated by the discretization of the time integral of the sub‐grid scale tensor. A new proposal for developing time‐accurate and congruent intermediate boundary conditions is addressed. Several tests for periodic and non‐periodic channel flows are presented. This study follows and completes the previous ones reported in (Int. J. Numer. Methods Fluids 2003; 42, 43 ). Copyright © 2005 John Wiley & Sons, Ltd.     

A Mach‐uniform preconditioner for incompressible and subsonic flows

In this study, a novel Mach‐uniform preconditioning method is developed for the solution of Euler equations at low subsonic and incompressible flow conditions. In contrast to the methods developed earlier in which the conservation of mass equation is preconditioned, in the present method, the conservation of energy equation is preconditioned, which enforces the divergence free constraint on the velocity field even at the limiting case of incompressible, zero Mach number flows. Despite most preconditioners, the proposed Mach‐uniform preconditioning method does not have a singularity point at zero Mach number. The preconditioned system of equations preserves the strong conservation form of Euler equations for compressible flows and recovers the artificial compressibility equations in the case of zero Mach number. A two‐dimensional Euler solver is developed for validation and performance evaluation of the present formulation for a wide range of Mach number flows. The validation cases studied show the convergence acceleration, stability, and accuracy of the present Mach‐uniform preconditioner in comparison to the non‐preconditioned compressible flow solutions. The convergence acceleration obtained with the present formulation is similar to those of the well‐known preconditioned system of equations for low subsonic flows and to those of the artificial compressibility method for incompressible flows. Copyright © 2013 John Wiley & Sons, Ltd.     

An immersed boundary vector potential-vorticity meshless solver of the incompressible Navier–Stokes equation

We present a strong form meshless solver for numerical solution of the nonstationary, incompressible, viscous Navier–Stokes equations in two (2D) and three dimensions (3D). We solve the flow equations in their stream function-vorticity (in 2D) and vector potential-vorticity (in 3D) formulation, by extending to 3D flows the boundary condition-enforced immersed boundary method, originally introduced in the literature for 2D problems. We use a Cartesian grid, uniform or locally refined, to discretize the spatial domain. We apply an explicit time integration scheme to update the transient vorticity equations, and we solve the Poisson type equation for the stream function or vector potential field using the meshless point collocation method. Spatial derivatives of the unknown field functions are computed using the discretization-corrected particle strength exchange method. We verify the accuracy of the proposed numerical scheme through commonly used benchmark and example problems. Excellent agreement with the data from the literature was achieved. The proposed method was shown to be very efficient, having relatively large critical time steps.     

Navier-Stokes equations with imposed pressure and velocity fluxes

Boundary value problems for Stokes and Navier-Stokes equations with non-standard boundary conditions are studied. Included is the case where the pressure or its normal derivative is given on some part of the boundary or the pressure is given up to a constant but given velocity flux. First, a variational formulation is introduced which is shown to be equivalent to the Stokes equations with the non-standard boundary conditions under consideration. The existence and uniqueness of the solution of the variational problem are studied. Secondly, most of the results obtained for the Stokes equations are extended to the case of the Navier-Stokes equations. The final section is devoted to numerical experiments, flows in pipes and physiological flows.     

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.     

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.