Two-stage fourth-order gas-kinetic scheme for three-dimensional Euler and Navier-Stokes solutions

ABSTRACT

For the one-stage third-order gas-kinetic scheme (GKS), successful applications have been achieved for the three-dimensional compressible flows [Pan, L., K. Xu, Q. Li, and J. Li. 2016. “An Efficient and Accurate Two-stage Fourth-order Gas-kinetic Scheme for the Navier-Stokes Equations.” Journal of Computational Physics 326: 197–221]. The high-order accuracy of the scheme is obtained by integrating a multidimensional time-accurate gas distribution function over the cell interface within a time step without using Gaussian quadrature points and Runge-Kutta time-stepping technique. However, to the further increase of the order of the scheme, such as the fourth-order one, the one step formulation becomes very complicated for the multidimensional flow. Recently, a two-stage fourth-order GKS with high efficiency has been constructed for two-dimensional inviscid and viscous flow computations ([Li, J., and Z. Du. 2016. “A Two-stage Fourth Order Time-accurate Discretization for Lax-Wendroff Type Flow Solvers I. Hyperbolic Conservation Laws.” SIAM Journal on Scientific Computing 38: 3046–3069]; Pan et al. 2016), and the scheme uses the time accurate flux function and its time derivatives. In this paper, a fourth-order GKS is developed for the three-dimensional flows under the two-stage framework. Based on the three-dimensional WENO reconstruction and flux evaluation at Gaussian quadrature points on a cell interface, the high-order accuracy in space is achieved first. Then, the two-stage time stepping method provides the high accuracy in time. In comparison with the formal third-order GKS [Pan, L., and K. Xu. 2015. “A Third-order Gas-kinetic Scheme for Three-dimensional Inviscid and Viscous Flow Computations.” Computers & Fluids 119: 250–260], the current fourth-order method not only improves the accuracy of the scheme, but also reduces the complexity of the gas-kinetic flux solver greatly. More importantly, the fourth-order GKS has the same robustness as the second-order shock capturing scheme [Xu, K. 2001. “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and its Connection with Artificial Dissipation and Godunov Method.” Journal of Computational Physics 171: 289–335]. Numerical results validate the outstanding reliability and applicability of the scheme for three-dimensional flows, such as the cases related to turbulent simulations.     

Nonlinear Galerkin method for the exterior nonstationary Navier-Stokes equations

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High‐order compact scheme for Boussinesq equations: implementation and numerical boundary condition issue

Application of the three‐point fourth‐order compact scheme to spatial differencing of the vorticity‐stream function‐density formulation of the two‐dimensional incompressible Boussinesq equations is presented. The details for the derivation of difference relations at boundaries to generate accurate and stable solutions are also given. To assess the numerical accuracy, two linear prototype test problems with known exact solution are used. The two‐dimensional planar and cylindrical lock‐exchange flow configurations are used to conduct the numerical experiments for the Boussinesq equations. Quantitative measures for the two linear prototype test problems and comparison of the results of this work with the published results for the planar lock‐exchange flow indicates the validity and accuracy of the three‐point fourth‐order compact scheme for numerical solution of two‐dimensional incompressible Boussinesq equations. In addition, the study of using different high‐order numerical boundary conditions for the implementation of the no‐penetration boundary condition for the density at no‐slip walls is considered. It is shown that the numerical solution is sensitive to the choice of difference relation for the density at boundaries and using an inappropriate difference relation leads to spurious numerical solution. Copyright © 2011 John Wiley & Sons, Ltd.     

A high‐order scheme for the incompressible Navier–Stokes equations with open boundary condition

We present a numerical scheme to solve the incompressible Navier–Stokes equations with open boundary condition. After replacing the incompressibility constraint by the pressure Poisson equation, the key is how to give an appropriate boundary condition for the pressure Poisson equation. We propose a new boundary condition for the pressure on the open boundary. Some numerical experiments are presented to verify the accuracy and stability of scheme. Copyright © 2013 John Wiley & Sons, Ltd.     

Robust high-order semi-implicit backward difference formulas for Navier-Stokes equations

Taylor-Hood finite elements provide a robust numerical discretization of Navier-Stokes equations (NSEs) with arbitrary high order of accuracy in space. To match the accuracy of the lowest degree Taylor-Hood element, we propose a very efficient time-stepping methods for unsteady flows, which are based on high-order semi-implicit backward difference formulas (SBDF) and the inclusion of grad -div term in the NSE. To mitigate the impact on the numerical accuracy (in time) of the extrapolation of the nonlinear term in SBDF, several variants of nonlinear extrapolation formulas are investigated. The first approach is based on an extrapolation of the nonlinear advection term itself. The second formula uses the extrapolation of the velocity prior to the evaluation of the nonlinear advection term as a whole. The third variant is constructed such that it produces similar error on both velocity and pressure to that with fully implicit backward difference formulas methods at a given order of accuracy. This can be achieved by fixing one-order higher than usually done in the extrapolation formula for the nonlinear advection term, while keeping the same extrapolation formula for the time derivative. The resulting truncation errors (in time) between these formulas are identified using Taylor expansions. These truncation error formulas are shown to properly represent the error seen in numerical tests using a 2D manufactured solution. Lastly, we show the robustness of the proposed semi-implicit methods by solving test cases with high Reynolds numbers using one of the nonlinear extrapolation formulas, namely, the 2D flow past circular cylinder at Re=300 and Re = 1000 and the 2D lid-driven cavity at Re = 50 000 and Re = 100 000. Our numerical solutions are found to be in a good agreement with those obtained in the literature, both qualitatively and quantitatively.     

不可压Navier-Stokes方程基于误差估算的网格自适应解法

首先导出了广义Stokes方程Petrov—Galerkin有限元数值解的当地事后误差估算公式;以非连续二阶鼓包（bump）函数空间为速度、压强误差的近似空间,该估算基于求解当地单元上的广义Stokes问题。然后,证明了误差估算值与精确误差之间的等价性。最后,将误差估算方法应用于Navier—Stokes环境,以进行不可压粘流计算中的网格自适应处理。数值实验中成功地捕获了多强度物理现象,验证了本文所发展的方法。     

A parallel two-level finite element method for the Navier-Stokes equations

Based on domain decomposition, a parallel two-level finite element method for the stationary Navier-Stokes equations is proposed and analyzed. The basic idea of the method is first to solve the Navier-Stokes equations on a coarse grid, then to solve the resulted residual equations in parallel on a fine grid. This method has low communication complexity. It can be implemented easily. By local a priori error estimate for finite element discretizations, error bounds of the approximate solution are derived. Numerical results are also given to illustrate the high efficiency of the method.     

Finite analytic boundary condition for a higher‐order finite analytic scheme

A higher‐order finite analytic scheme based on one‐dimensional finite analytic solutions is used to discretize three‐dimensional equations governing turbulent incompressible free surface flow. In order to preserve the accuracy of the numerical scheme, a new, finite analytic boundary condition is proposed for an accurate numerical solution of the partial differential equation. This condition has higher‐order accuracy. Thus, the same order of accuracy is used for the boundary. Boundary conditions were formulated and derived for fluid inflow, outflow, impermeable surfaces and symmetry planes. The derived boundary conditions are treated implicitly and updated with the solution of the problem. The basic idea for the derivation of boundary conditions was to use the discretized form of the governing equations for the fluid flow simplified on the boundaries and flow information. To illustrate the influence of the higher‐order effects at the boundaries, another, lower‐order finite analytic boundary condition, is suggested. The simulations are performed to demonstrate the validity of the present scheme and boundary conditions for a Wigley hull advancing in calm water. Copyright © 2005 John Wiley & Sons, Ltd.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid: 3D validation

A three‐dimensional finite element method for incompressible multiphase flows with capillary interfaces is developed based on a (formally) second‐order projection scheme. The discretization is on a fixed (Eulerian) reference grid with an edge‐based local h‐refinement in the neighbourhood of the interfaces. The fluid phases are identified and advected using the level‐set function. The reference grid is then temporarily reconnected around the interface to maintain optimal interpolations accounting for the singularities of the primary variables. Using a time splitting procedure, the convection substep is integrated with an explicit scheme. The remaining generalized Stokes problem is solved by means of a pressure‐stabilized projection. This method is simple and efficient, as demonstrated by a wide range of difficult free‐surface validation problems, considered in the paper. Copyright © 2005 John Wiley & Sons, Ltd.     

A new full discrete stabilized viscosity method for transient Navier-Stokes equations

A new full discrete stabilized viscosity method for the transient Navier-Stokes equations with the high Reynolds number （small viscosity coefficient） is proposed based on the pressure projection and the extrapolated trapezoidal rule. The transient Navier-Stokes equations are fully discretized by the continuous equal-order finite elements in space and the reduced Crank-Nicolson scheme in time. The new stabilized method is stable and has many attractive properties. First, the system is stable for the equal-order combination of discrete continuous velocity and pressure spaces because of adding a pres- sure projection term. Second, the artifical viscosity parameter is added to the viscosity coefficient as a stability factor, so the system is antidiffusive. Finally, the method requires only the solution to a linear system at every time step. Stability and convergence of the method is proved. The error estimation results show that the method has a second-order accuracy, and the constant in the estimation is independent of the viscosity coefficient. The numerical results are given, which demonstrate the advantages of the method presented.     

A projection scheme for incompressible multiphase flow using adaptive Eulerian grid

This paper presents a finite element method for incompressible multiphase flows with capillary interfaces based on a (formally) second‐order projection scheme. The discretization is on a fixed Eulerian grid. The fluid phases are identified and advected using a level set function. The grid is temporarily adapted around the interfaces in order to maintain optimal interpolations accounting for the pressure jump and the discontinuity of the normal velocity derivatives. The least‐squares method for computing the curvature is used, combined with piecewise linear approximation to the interface. The time integration is based on a formally second order splitting scheme. The convection substep is integrated over an Eulerian grid using an explicit scheme. The remaining generalized Stokes problem is solved by means of a formally second order pressure‐stabilized projection scheme. The pressure boundary condition on the free interface is imposed in a strong form (pointwise) at the pressure‐computation substep. This allows capturing significant pressure jumps across the interface without creating spurious instabilities. This method is simple and efficient, as demonstrated by the numerical experiments on a wide range of free‐surface problems. Copyright © 2004 John Wiley & Sons, Ltd.     

Local and parallel finite element algorithms based on two-grid discretization for steady Navier-Stokes equations

Local and parallel finite element algorithms based on two-grid discretization for Navier-Stokes equations in two dimension are presented. Its basis is a coarse finite element space on the global domain and a fine finite element space on the subdomain. The local algorithm consists of finding a solution for a given nonlinear problem in the coarse finite element space and a solution for a linear problem in the fine finite element space, then droping the coarse solution of the region near the boundary. By overlapping domain decomposition, the parallel algorithms are obtained. This paper analyzes the error of these algorithms and gets some error estimates which are better than those of the standard finite element method. The numerical experiments are given too. By analyzing and comparing these results, it is shown that these algorithms are correct and high efficient.     

A boundary element formulation for natural convection problems

This paper presents a boundary element formulation employing a penalty function technique for two-dimensional steady thermal convection problems. By regarding the convective and buoyancy force terms in Navier-Stokes equations as body forces, the standard elastostatics analysis can be extended to solve the Navier-Stokes equations. In a similar manner, the standard potential analysis is extended to solve the energy transport equation. Finally, some numerical results are included, for typical natural convection problems, in order to demonstrate the efficiency of the present method.     

A high-order characteristics/finite element method for the incompressible Navier-Stokes equations

In this paper we consider a discretization of the incompressible Navier-Stokes equations involving a second-order time scheme based on the characteristics method and a spatial discretization of finite element type. Theoretical and numerical analyses are detailed and we obtain stability results abnd optimal eror estimates on the velocity and pressure under a time step restriction less stringent than the standard Courant-Freidrichs-Levy condition. Finally, some numerical results obtained wiht the code N3S are shown which justify the interest of this scheme and its advantages with respect to an analogous first-order time scheme. © 1997 John Wiley & Sons, Ltd.     

A streamline diffusion nonconforming finite element method for the time-dependent linearized Navier-Stokes equations

A nonconforming finite element method of finite difference streamline diffusion type is proposed to solve the time-dependent linearized Navier-Stokes equations. The backward Euler scheme is used for time discretization. Crouzeix-Raviart nonconforming finite element approximation, namely, nonconforming （P1）2 - P0 element, is used for the velocity and pressure fields with the streamline diffusion technique to cope with usual instabilities caused by the convection and time terms. Stability and error estimates are derived with suitable norms.     

A compact fourth-order finite difference scheme for the steady incompressible Navier-Stokes equations

We note in this study that the Navier-Stokes equations, when expressed in streamfunction-vorticity form, can be approximated to fourth-order accuracy with stencils extending only over a 3 x 3 square of points. The key advantage of the new compact fourth-order scheme is that it allows direct iteration for low-to-medium Reynolds numbers. Numerical solutions are obtained for the model problem of the driven cavity and compared with solutions available in the literature. For Re ? 7500 point-SOR iteration is used and the convergence is fast.     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases. The project supported by the China NKBRSF (2001CB409604) The English text was polished by Yunming Chen     

A total linearization method for solving viscous free boundary flow problems by the finite element method

In this paper a total linearization method is derived for solving steady viscous free boundary flow problems (including capillary effects) by the finite element method. It is shown that the influence of the geometrical unknown in the totally linearized weak formulation can be expressed in terms of boundary integrals. This means that the implementation of the method is simple. Numerical experiments show that the iterative method gives accurate results and converges very fast.     

Analysis of electric boundary condition effects on crack propagation in piezoelectric ceramics

There are three types of cracks: impermeable crack, permeable crack and conducting crack, with different electric boundary conditions on faces of cracks in piezoelectric ceramics, which poses difficulties in the analysis of piezoelectric fracture problems. In this paper, in contrast to our previous FEM formulation, the numerical analysis is based on the used of exact electric boundary conditions at the crack faces, thus the common assumption of electric impermeability in the FEM analysis is avoided. The crack behavior and elasto-electric fields near a crack tip in a PZT-5 piezoelectric ceramic under mechanical, electrical and coupled mechanical-electrical loads with different electric boundary conditions on crack faces are investigated. It is found that the dielectric medium between the crack faces will reduce the singularity of stress and electric displacement. Furthermore, when the permittivity of the dielectric medium in the crack gap is of the same order as that of the piezoelectric ceramic, the crack becomes a conducting crack, the applied electric field has no effect on the crack propagation. The project supported by the National Natural Science Foundation of China (19672026, 19891180)