一类三维时空二阶精度高分辨率MmB差分格式

直接把Nessyahu和Tadmor^[1,2]的思想推广到三维非线性双曲型守恒律情形，以交错形式Lax—Friedrichs格式为基本模块，使用二阶分片线性逼近代替一阶分片常数逼近，减少了Lax—Friedrichs格式的过多数值粘性，通过对混合导数离散形式的适当处理，构造了一类不须解Riemann问题、具有时空二阶精度高分辨率的MmB差分格式。这些差分格式很容易推广到向量系统中去。最后，一些数值模拟计算结果也证明了这些差分格式的有效性。     

Transient shock wave flows in tubes with a sudden change in cross section

This paper describes propagation of shock waves within circular cross-section shock tubes with a sudden area change in cross section. A dispersion-controlled scheme was used to solve the Euler equations assuming axisymmetric flows. For experimental visualizations an aspheric cylindrical test section was designed to keep collimated incident light rays parallel once they were reflected or refracted on the inner and outer surfaces of the test section. For effective comparisons with experimental results, equivalent numerical interferograms were constructed to demonstrate effectiveness of the numerical method and verify the observed shock-wave phenomena. The numerical method was used to calculate three further cases with variations of the initial shock-wave Mach number and the flow geometry to clarify the role of these parameters. Complex transient shock-wave phenomena, such as shock-wave reflection, shock/vortex interaction and shock-wave focusing were observed in these cases, and interpreted with shock wave theory. In addition, the research clearly shows that combination of CFD with experiments is effective to highlight physical phenomena in axisymmetric flows. Received 15 June 1996 / Accepted 20 December 1996     

Assessment of WENO schemes for multi‐dimensional Euler equations using GPU

This paper presents a detailed study on the implementation of Weighted Essentially Non‐Oscillatory (WENO) schemes on GPU. GPU implementation of up to ninth‐order accurate WENO schemes for the multi‐dimensional Euler equations of gas dynamics is presented. The implementation detail is discussed in the paper. The computational times of different schemes are obtained and the speedups are reported for different number of grid points. Furthermore, the execution times for the main kernels of the code are given and compared with each other. The numerical experiments show the speedups for the WENO schemes are very promising especially for fine grids. Copyright © 2014 John Wiley & Sons, Ltd.     

Equivalence conditions between linear Lagrangian finite element and node‐centred finite volume schemes for conservation laws in cylindrical coordinates

A complete set of equivalence conditions, relating the mass‐lumped Bubnov–Galerkin finite element (FE) scheme for Lagrangian linear elements to node‐centred finite volume (FV) schemes, is derived for the first time for conservation laws in a three‐dimensional cylindrical reference. Equivalence conditions are used to devise a class of FV schemes, in which all grid‐dependent quantities are defined in terms of FE integrals. Moreover, all relevant differential operators in the FV framework are consistent with their FE counterparts, thus opening the way to the development of consistent hybrid FV/FE schemes for conservation laws in a three‐dimensional cylindrical coordinate system. The two‐dimensional schemes for the polar and the axisymmetrical cases are also explicitly derived. Numerical experiments for compressible unsteady flows, including expanding and converging shock problems and the interaction of a converging shock waves with obstacles, are carried out using the new approach. The results agree fairly well with one‐dimensional simulations and simplified models. Copyright © 2013 John Wiley & Sons, Ltd.     

Numerical simulation of a single bubble by compressible two‐phase fluids

The present work deals with the numerical investigation of a collapsing bubble in a liquid–gas fluid, which is modeled as a single compressible medium. The medium is characterized by the stiffened gas law using different material parameters for the two phases. For the discretization of the stiffened gas model, the approach of Saurel and Abgrall is employed where the flow equations, here the Euler equations, for the conserved quantities are approximated by a finite volume scheme, and an upwind discretization is used for the non‐conservative transport equations of the pressure law coefficients. The original first‐order discretization is extended to higher order applying second‐order ENO reconstruction to the primitive variables. The derivation of the non‐conservative upwind discretization for the phase indicator, here the gas fraction, is presented for arbitrary unstructured grids. The efficiency of the numerical scheme is significantly improved by employing local grid adaptation. For this purpose, multiscale‐based grid adaptation is used in combination with a multilevel time stepping strategy to avoid small time steps for coarse cells. The resulting numerical scheme is then applied to the numerical investigation of the 2‐D axisymmetric collapse of a gas bubble in a free flow field and near to a rigid wall. The numerical investigation predicts physical features such as bubble collapse, bubble splitting and the formation of a liquid jet that can be observed in experiments with laser‐induced cavitation bubbles. Opposite to the experiments, the computations reveal insight to the state inside the bubble clearly indicating that these features are caused by the acceleration of the gas due to shock wave focusing and reflection as well as wave interaction processes. While incompressible models have been used to provide useful predictions on the change of the bubble shape of a collapsing bubble near a solid boundary, we wish to study the effects of shock wave emissions into the ambient liquid on the bubble collapse, a phenomenon that may not be captured using an incompressible fluid model. Copyright © 2009 John Wiley & Sons, Ltd.     

Unstructured finite volume discretization of two‐dimensional depth‐averaged shallow water equations with porosity

This paper deals with the numerical discretization of two‐dimensional depth‐averaged models with porosity. The equations solved by these models are similar to the classic shallow water equations, but include additional terms to account for the effect of small‐scale impervious obstructions which are not resolved by the numerical mesh because their size is smaller or similar to the average mesh size. These small‐scale obstructions diminish the available storage volume on a given region, reduce the effective cross section for the water to flow, and increase the head losses due to additional drag forces and turbulence. In shallow water models with porosity these effects are modelled introducing an effective porosity parameter in the mass and momentum conservation equations, and including an additional drag source term in the momentum equations. This paper presents and compares two different numerical discretizations for the two‐dimensional shallow water equations with porosity, both of them are high‐order schemes. The numerical schemes proposed are well‐balanced, in the sense that they preserve naturally the exact hydrostatic solution without the need of high‐order corrections in the source terms. At the same time they are able to deal accurately with regions of zero porosity, where the water cannot flow. Several numerical test cases are used in order to verify the properties of the discretization schemes proposed. Copyright © 2009 John Wiley & Sons, Ltd.     

Zur Elektronenstruktur metallorganischer Komplexe der f‐Elemente. 67 Erstmalige parametrische Analyse des Absorptionsspektrums einer Molekülverbindung des CeIIIμ: Tris(η5‐tetramethylcyclopentadienyl)cer

Electronic Structures of Organometallic Complexes of f Elements. 67 First Parametric Analysis of the Absorption Spectrum of a Molecular Compound of Ce^IIIμ: Tris(η⁵‐tetramethylcyclopentadienyl)cerium(III) The absorption spectra (in the IR/NIR/Vis/UV range) of Ce(C₅Me₄H)₃ ( 1 ) and La(C₅Me₄H)₃ ( 2 ) were recorded at room and low temperatures. From the spectra obtained, two alternative closely related crystal field (CF) splitting patterns of 1 could be derived, and simulated by fitting the free parameters of a phenomenological Hamiltonian. The fact that the difference of the experimental energies of the barycenters of CF levels of the multiplets ²F_7/2 and ²F_5/2 is larger than in the gaseous free Ce³⁺ ion (“anti”‐relativistic nephelauxetic effect) could be explained by coupling effects of these multiplets via the CF, resulting in lower spin‐orbit coupling parameters than in the case of the gaseous free Ce³⁺ ion. The experimentally derived CF splitting pattern of 1 is compared with the predictions of previous non‐relativistic SW‐Xα and relativistic DV‐Xα calculations.     

High-order non-uniform grid schemes for numerical simulation of hypersonic boundary-layer stability and transition

The direct numerical simulation of receptivity, instability and transition of hypersonic boundary layers requires high-order accurate schemes because lower-order schemes do not have an adequate accuracy level to compute the large range of time and length scales in such flow fields. The main limiting factor in the application of high-order schemes to practical boundary-layer flow problems is the numerical instability of high-order boundary closure schemes on the wall. This paper presents a family of high-order non-uniform grid finite difference schemes with stable boundary closures for the direct numerical simulation of hypersonic boundary-layer transition. By using an appropriate grid stretching, and clustering grid points near the boundary, high-order schemes with stable boundary closures can be obtained. The order of the schemes ranges from first-order at the lowest, to the global spectral collocation method at the highest. The accuracy and stability of the new high-order numerical schemes is tested by numerical simulations of the linear wave equation and two-dimensional incompressible flat plate boundary layer flows. The high-order non-uniform-grid schemes (up to the 11th-order) are subsequently applied for the simulation of the receptivity of a hypersonic boundary layer to free stream disturbances over a blunt leading edge. The steady and unsteady results show that the new high-order schemes are stable and are able to produce high accuracy for computations of the nonlinear two-dimensional Navier–Stokes equations for the wall bounded supersonic flow.     

Application of a fourth-order relaxation scheme to hyperbolic systems of conservation laws

A fourth-order relaxation scheme is derived and applied to hyperbolic systems of conservation laws in one and two space dimensions. The scheme is based on a fourth-order central weighted essentially nonoscillatory (CWENO) reconstruction for one-dimensional cases, which is generalized to two-dimensional cases by the dimension-by-dimension approach. The large stability domain Runge-Kutta-type solver ROCK4 is used for time integration. The resulting method requires neither the use of Riemann solvers nor the computation of Jacobians and therefore it enjoys the main advantage of the relaxation schemes. The high accuracy and high-resolution properties of the present method are demonstrated in one- and two-dimensional numerical experiments. The project supported by the National Natural Science Foundation of China (60134010) The English text was polished by Yunming Chen.