An adaptive GRP scheme for compressible fluid flows

This paper presents a second-order accurate adaptive generalized Riemann problem (GRP) scheme for one and two dimensional compressible fluid flows. The current scheme consists of two independent parts: Mesh redistribution and PDE evolution. The first part is an iterative procedure. In each iteration, mesh points are first redistributed, and then a conservative interpolation formula is used to calculate the cell-averages and the slopes of conservative variables on the resulting new mesh. The second part is to evolve the compressible fluid flows on a fixed nonuniform mesh with the Eulerian GRP scheme, which is directly extended to two-dimensional arbitrary quadrilateral meshes. Several numerical examples show that the current adaptive GRP scheme does not only improve the resolution as well as accuracy of numerical solutions with a few mesh points, but also reduces possible errors or oscillations effectively.     

Conservative form of interpolated differential operator scheme for compressible and incompressible fluid dynamics

The proposed scheme, which is a conservative form of the interpolated differential operator scheme (IDO-CF), can provide high accurate solutions for both compressible and incompressible fluid equations. Spatial discretizations with fourth-order accuracy are derived from interpolation functions locally constructed by both cell-integrated values and point values. These values are coupled and time-integrated by solving fluid equations in the flux forms for the cell-integrated values and in the derivative forms for the point values. The IDO-CF scheme exactly conserves mass, momentum, and energy, retaining the high resolution more than the non-conservative form of the IDO scheme. A direct numerical simulation of turbulence is carried out with comparable accuracy to that of spectral methods. Benchmark tests of Riemann problems and lid-driven cavity flows show that the IDO-CF scheme is immensely promising in compressible and incompressible fluid dynamics studies.     

The space–time CESE method for solving special relativistic hydrodynamic equations

The special relativistic hydrodynamic equations are more complicated than the classical ones due to the nonlinear and implicit relations that exist between conservative and primitive variables. In this article, a space–time conservation element and solution element (CESE) method is proposed for solving these equations in one and two space dimensions. The CESE method has capability to capture sharp propagating wavefront of the relativistic fluids without excessive numerical diffusion or spurious oscillations. In contrast to the existing upwind finite volume schemes, the Riemann solver and reconstruction procedure are not the building blocks of the suggested method. The method differs from previous techniques because of global and local flux conservation in a space–time domain without resorting to interpolation or extrapolation. The scheme is efficient, robust, and gives results comparable to those obtained with more sophisticated algorithms, even in highly relativistic two-dimensional test problems.     

A high-order cell-centered Lagrangian scheme for two-dimensional compressible fluid flows on unstructured meshes

We present a high-order cell-centered Lagrangian scheme for solving the two-dimensional gas dynamics equations on unstructured meshes. A node-based discretization of the numerical fluxes for the physical conservation laws allows to derive a scheme that is compatible with the geometric conservation law (GCL). Fluxes are computed using a nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The first-order scheme is conservative for momentum and total energy, and satisfies a local entropy inequality in its semi-discrete form. The two-dimensional high-order extension is constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess this new scheme. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new scheme.     

采用非交错网格压力修正法计算不可压及可压亚、跨、超音速流动

采用非交错网格压力修正法计算不可压及可压亚、跨、超音速流动朱斌，蔡睿贤，蒋洪德（中国科学院工程热物理研究所北京１０００８０）关键词：压力修正法，亚、跨、超音速流动一、前言源于计算不可压流动的压力修正方法，由于具有计算时间短及不受马赫数限制的优点，近年...     

多介质流体力学计算的守恒型高分辨率格式

应用Lagrange坐标系下的守恒型格式计算多介质流体力学问题,在物质交界面附近采用一阶格式的数值通量,而在其余部分采用高分辨率格式的数值通量,不仅保持了高分辨率的良好性质,而且消除了一般的守恒型格式在界面附近所产生的震荡.     

A high-order cell-centered Lagrangian scheme for compressible fluid flows in two-dimensional cylindrical geometry

The goal of this paper is to present high-order cell-centered schemes for solving the equations of Lagrangian gas dynamics written in cylindrical geometry. A node-based discretization of the numerical fluxes is obtained through the computation of the time rate of change of the cell volume. It allows to derive finite volume numerical schemes that are compatible with the geometric conservation law (GCL). Two discretizations of the momentum equations are proposed depending on the form of the discrete gradient operator. The first one corresponds to the control volume scheme while the second one corresponds to the so-called area-weighted scheme. Both formulations share the same discretization for the total energy equation. In both schemes, fluxes are computed using the same nodal solver which can be viewed as a two-dimensional extension of an approximate Riemann solver. The control volume scheme is conservative for momentum, total energy and satisfies a local entropy inequality in its first-order semi-discrete form. However, it does not preserve spherical symmetry. On the other hand, the area-weighted scheme is conservative for total energy and preserves spherical symmetry for one-dimensional spherical flow on equi-angular polar grid. The two-dimensional high-order extensions of these two schemes are constructed employing the generalized Riemann problem (GRP) in the acoustic approximation. Many numerical tests are presented in order to assess these new schemes. The results obtained for various representative configurations of one and two-dimensional compressible fluid flows show the robustness and the accuracy of our new schemes.     

Binary and ternary oscillations in a cubic numerical scheme

We study a central difference semi-discretization of the cubic scalar conservation law. Both spatial period-2 (binary) and period-3 (ternary) oscillations are stationary solutions of this scheme, and we find where each type is linearly stable. We observe numerically the formation of ternary oscillations, to the left of Riemann shock initial data with u_r = 0, while binary oscillations form to the right of Riemann rarefaction data having u_l = 0. We derive modulation equations for both wave patterns, using them to show that binary oscillations generated by the scheme are numerical artifacts, while computing an explicit solution for the ternary modulation equations. For positive initial data, the ternary modulation equations remain hyperbolic, while the solutions enter an elliptic region for data of both signs. Conditions under which solutions of the ternary modulation equations can be inverted to yield period-3 oscillations are also discussed.     

Skew-symmetric form of convective terms and fully conservative finite difference schemes for variable density low-Mach number flows

The form of convective terms for compressible flow equations is discussed in the same way as for an incompressible one by Morinishi et al. [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, J. Comput. Phys. 124 (1998) 90], and fully conservative finite difference schemes suitable for shock-free unsteady compressible flow simulations are proposed. Commutable divergence, advective, and skew-symmetric forms of convective terms are defined by including the temporal derivative term for compressible flow. These forms are analytically equivalent if the continuity is satisfied, and the skew-symmetric form is secondary conservative without the aid of the continuity, while the divergence form is primary conservative. The relations between the present and existing quasi-skew-symmetric forms are also revealed. Commutable fully discrete finite difference schemes of convection are then derived in a staggered grid system, and they are fully conservative provided that the corresponding discrete continuity is satisfied. In addition, a semi-discrete convection scheme suitable for compact finite difference is presented based on the skew-symmetric form. The conservation properties of the present schemes are demonstrated numerically in a three-dimensional periodic inviscid flow. The proposed fully discrete fully conservative second-order accurate scheme is also used to perform the DNS of compressible isotropic turbulence and the simulation of open cavity flow.     

一维多介质可压缩流动数值方法

应用高精度界面追踪方法计算一般状态方程的多介质可压缩流动问题;应用LevelSet技术捕捉界面位置,在界面附近采用守恒数值离散,用双波近似求解一般状态方程Riemann问题,并采用统一高阶PPM格式进行内点和交界面点的计算.一维算例表明,该方法对于光滑区域以及多介质交界面具有二阶精度,能准确地模拟交界面的位置,交界面计算无数值振荡和数值耗散,并能处理一般状态方程的多介质可压缩流动问题.     

Preventing numerical errors generated by interface-capturing schemes in compressible multi-material flows

In the present work, errors generated in computations of compressible multi-material flows using shock-capturing schemes are examined, specifically pressure oscillations (when the specific heats ratio is variable), but also temperature spikes and species conservation errors. These numerical errors are generated at material discontinuities due to an inconsistent treatment of the convective terms. Though temperature errors are irrelevant to solutions to the Euler equations, it is shown that they have the potential to lead to problems when physical diffusion is included, i.e., for the Navier–Stokes equations. These errors are studied analytically and numerically by considering the one-dimensional advection of isolated material discontinuities. A methodology preventing such errors for weighted essentially non-oscillatory (WENO) schemes is presented, in which modified WENO weights are used to solve the transport equation for mass fraction in conservative form to prevent temperature and species conservation errors. Pressure errors are prevented by solving an additional transport equation for a given function of the ratio of specific heats. Several multi-dimensional problems with various discontinuities (shocks, material interfaces and contact discontinuities), including the single-mode Richtmyer–Meshkov instability, and turbulence are considered to test the method.     

An unconditionally stable fully conservative semi-Lagrangian method

Semi-Lagrangian methods have been around for some time, dating back at least to [3]. Researchers have worked to increase their accuracy, and these schemes have gained newfound interest with the recent widespread use of adaptive grids where the CFL-based time step restriction of the smallest cell can be overwhelming. Since these schemes are based on characteristic tracing and interpolation, they do not readily lend themselves to a fully conservative implementation. However, we propose a novel technique that applies a conservative limiter to the typical semi-Lagrangian interpolation step in order to guarantee that the amount of the conservative quantity does not increase during this advection. In addition, we propose a new second step that forward advects any of the conserved quantity that was not accounted for in the typical semi-Lagrangian advection. We show that this new scheme can be used to conserve both mass and momentum for incompressible flows. For incompressible flows, we further explore properly conserving kinetic energy during the advection step, but note that the divergence free projection results in a velocity field which is inconsistent with conservation of kinetic energy (even for inviscid flows where it should be conserved). For compressible flows, we rely on a recently proposed splitting technique that eliminates the acoustic CFL time step restriction via an incompressible-style pressure solve. Then our new method can be applied to conservatively advect mass, momentum and total energy in order to exactly conserve these quantities, and remove the remaining time step restriction based on fluid velocity that the original scheme still had.     

Self-adjusting entropy-stable scheme for compressible Euler equations

In this work,a self-adjusting entropy-stable scheme is proposed for solving compressible Euler equations.The entropy-stable scheme is constructed by combining the entropy conservative flux with a suitable diffusion operator.The entropy has to be preserved in smooth solutions and be dissipated at shocks.To achieve this,a switch function,which is based on entropy variables,is employed to make the numerical diffusion term be automatically added around discontinuities.The resulting scheme is still entropy-stable.A number of numerical experiments illustrating the robustness and accuracy of the scheme are presented.From these numerical results,we observe a remarkable gain in accuracy.     

Completely Conservative and Oscillationless Semi-Lagrangian Schemes for Advection Transportation

In this paper, we present a new type of semi-Lagrangian scheme for advection transportation equation. The interpolation function is based on a cubic polynomial and is constructed under the constraints of conservation of cell-integrated average and the slope modification. The cell-integrated average is defined via the spatial integration of the interpolation function over a single grid cell and is advanced using a flux form. Nonoscillatory interpolation is constructed by choosing proper approximation to the cell-center values of the first derivative of the interpolation function, which appears to be a free parameter in the present formulation. The resulting scheme is exactly conservative regarding the cell average of the advected quantity and does not produce any spurious oscillation. Oscillationless solutions to linear transportation problems were obtained. Incorporated with an entropy-enforcing numerical flux, the presented schemes can accurately compute shocks and sonic rarefaction waves when applied to nonlinear problems.     

Efficient kinetic schemes for steady and unsteady flow simulations on unstructured meshes

This paper presents efficient second-order kinetic schemes on unstructured meshes for both compressible unsteady and incompressible steady flows. For compressible unsteady flows, a time-dependent gas distribution function with a discontinuous particle velocity space at a cell interface is constructed and used for the evaluations of both numerical fluxes and conservative flow variables. As a result, a compact scheme on the unstructured meshes is developed. For incompressible steady flows, a continuous second-order gas-kinetic BGK type scheme is presented, for which the time-dependent gas distribution function with a continuous particle velocity is used on unstructured meshes. The efficiency of the schemes lies in the fact that the slopes of the flow variables inside each cell can be constructed using values of the flow variables within that cell only without involving neighboring cells. Therefore, even with the stencil of a first-order scheme, a high resolution method is constructed. Numerical examples are presented which are compared with the benchmark solutions and the experimental measurements.     

双曲型守恒律的一种高精度TVD差分格式

构造了一维双曲型守恒律方程的一个高精度高分辨率的守恒型TVD差分格式.其主要思想是:首先将计算区域划分为互不重叠的小单元,且每个小单元再根据希望的精度阶数分为细小单元;其次,根据流动方向将通量分裂为正、负通量,并通过小单元上的高阶插值逼近得到了细小单元边界上的正、负数值通量,为避免由高阶插值产生的数值振荡,进一步根据流向对其进行TVD校正;再利用高阶Runge KuttaTVD离散方法对时间进行离散,得到了高阶全离散方法.进一步推广到一维方程组情形.最后对一维欧拉方程组计算了几个算例.     

Failsafe flux limiting and constrained data projections for equations of gas dynamics

A new approach to flux limiting for systems of conservation laws is presented. The Galerkin finite element discretization/L² projection is equipped with a failsafe mechanism that prevents the birth and growth of spurious local extrema. Within the framework of a synchronized flux-corrected transport (FCT) algorithm, the velocity and pressure fields are constrained using node-by-node transformations from the conservative to the primitive variables. An additional correction step is included to ensure that all the quantities of interest (density, velocity, pressure) are bounded by the physically admissible low-order values. The result is a conservative and bounded scheme with low numerical diffusion. The new failsafe FCT limiter is integrated into a high-resolution finite element scheme for the Euler equations of gas dynamics. Also, bounded L² projection operators for conservative interpolation/initialization are designed. The performance of the proposed limiting strategy and the need for a posteriori control of flux-corrected solutions are illustrated by numerical examples.     

A mass,energy, vorticity,and potential enstrophy conserving lateral boundary scheme for the shallow water equations using piecewise linear boundary approximations

A numerical scheme for treating fluid–land boundaries in inviscid shallow water flows is derived that approximates boundary profiles with piecewise linear segments (shaved cells) while conserving the domain-summed mass, energy, vorticity, and potential enstrophy. The new scheme is a generalization of a previous scheme that also conserves these quantities but uses stairsteps to approximate boundary profiles. Numerical simulations are carried out demonstrating the conservation properties and accuracy of the piecewise linear boundary scheme (the PLS) for inviscid flows and comparing its performance with that of the stairstep scheme (the STS). It is found that while both schemes conserve all four domain-summed quantities, the PLS generates depth and velocity fields that are one-half to one order more accurate than those generated by the STS, and it generates vorticity and potential vorticity fields that are at least as accurate as those generated by the STS and often more accurate. The higher accuracy of the PLS is due to its ability to generate smoother flow fields near boundaries of arbitrary shape.     

关于分区模拟无粘流内边界守恒算法的相容性

本文讨论用分区方法计算可压缩无粘流动的内边界通量守恒算法。分析指出插值型通量守恒算法不保证相容性条件,计算表明这种不相容性能引起数值解出现明显误差,或破坏其稳定性、收敛性。为此,文中提出内边界采用自适应算法并给出了算例。     

A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh

The consistent and conservative scheme developed on a rectangular collocated mesh [M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system, Journal of Computational Physics 227 (2007) 174–204] and on an arbitrary collocated mesh [M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh, Journal of Computational Physics 227 (2007) 205–228] has been extended and specially designed for calculation of the Lorentz force on a staggered grid system (Part III) by solving the electrical potential equation for magnetohydrodynamics (MHD) at a low magnetic Reynolds number. In a staggered mesh, pressure (p) and electrical potential (φ) are located in the cell center, while velocities and current fluxes are located on the cell faces of a main control volume. The scheme numerically meets the physical conservation laws, charge conservation law and momentum conservation law. Physically, the Lorentz force conserves the momentum when the magnetic field is constant or spatial coordinate independent. The calculation of current density fluxes on cell faces is conducted using a scheme consistent with the discretization for solution of the electrical potential Poisson equation, which can ensure the calculated current density conserves the charge. A divergence formula of the Lorentz force is used to calculate the Lorentz force at the cell center of a main control volume, which can numerically conserve the momentum at constant or spatial coordinate independent magnetic field. The calculated cell-center Lorentz forces are then interpolated to the cell faces, which are used to obtain the corresponding velocity fluxes by solving the momentum equations. The “conservative” is an important property of the scheme, which can guarantee computational accuracy of MHD flows at high Hartmann number with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers. 2D fully developed MHD flows with analytical solutions available have been conducted to validate the scheme at a staggered mesh. 3D MHD flows, with the experimental data available, at a constant magnetic field in a rectangular duct with sudden expansion and at a varying magnetic field in a rectangular duct are conducted on a staggered mesh to verify the computational accuracy of the scheme. It is expected that the scheme for the Lorentz force can be employed together with a fully conservative scheme for the convective term and the pressure term [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, Journal of Computational Physics 143 (1998) 90–124] for direct simulation of MHD turbulence and MHD instability with good accuracy at a staggered mesh.