Improved multislope MUSCL reconstruction on unstructured grids for shallow water equations

In shallow water flow and transport modeling, the monotonic upstream‐centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second‐order accuracy. The most important step in MUSCL‐type schemes is MUSCL reconstruction, which calculate‐extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2‐point stencil that uses the cell centers of the neighbor cell and the so‐called far‐neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.     

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the structured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the solution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.     

A bore-capturing finite volume method for open-channel flows

A high-resolution finite volume hydrodynamic solver is presented for open-channel flows based on the 2D shallow water equations. This Godunov-type upwind scheme uses an efficient Harten–Lax–van Leer (HLL) approximate Riemann solver capable of capturing bore waves and simulating supercritical flows. Second-order accuracy is achieved by means of MUSCL reconstruction in conjunction with a Hancock two-stage scheme for the time integration. By using a finite volume approach, the computational grid can be irregular which allows for easy boundary fitting. The method can be applied directly to model 1D flows in an open channel with a rectangular cross-section without the need to modify the scheme. Such a modification is normally required for solving the 1D St Venant equations to take account of the variation of channel width. The numerical scheme and results of three test problems are presented in this paper. © 1998 John Wiley & Sons, Ltd.     

用隐式方法和WENO格式计算悬停旋翼跨声速无粘流场

寻找一种能够准确计算以涡为主要特征的复杂流场和克服尾迹耗散问题的数值方法,一直是旋翼空气动力学研究的热点和难点。本文发展了一种基于高阶迎风格式计算悬停旋翼无粘流场的隐式数值方法。无粘通量采用Roe通量差分分裂格式,为提高精度,使用五阶WENO格式进行左右状态插值,并与MUSCL插值进行比较。为提高收敛到定常解的效率,时间推进采用LU-SGS隐式方法。用该方法对一跨声速悬停旋翼无粘流场进行了数值计算,数值结果表明WENO-Roe的激波分辨率高于MUSCL-Roe,体现出了格式精度的提高对计算结果的改善,LU-SGS隐式方法的计算效率比5步Runge-Kutta显式方法的高。     

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boun...     

A pressure correction‐volume of fluid method for simulation of two‐phase flows

A pressure correction method coupled with the volume of fluid (VOF) method is developed to simulate two‐phase flows. A volume fraction function is introduced in the VOF method and is governed by an advection equation. A modified monotone upwind scheme for a conservation law (modified MUSCL) is used to solve the solution of the advection equation. To keep the initial sharpness of an interface, a slope modification scheme is introduced. The continuum surface tension (CST) model is used to calculate the surface tension force. Three schemes, central‐upwind, Parker–Youngs, and mixed schemes, are introduced to compute the interface normal vector and the gradient of the volume fraction function. Moreover, a height function technique is applied to compute the local curvature of the interface. Several basic test problems are performed to check the order of accuracy of the present numerical schemes for computing the interface normal vector and the gradient of the volume fraction function. Three physical problems, two‐dimensional broken dam problem, static drop, and spurious currents, and three‐dimensional rising bubble, are performed to demonstrate the efficiency and accuracy of the pressure correction method. Copyright © 2010 John Wiley & Sons, Ltd.     

Verification of Euler/Navier–Stokes codes using the method of manufactured solutions

The method of manufactured solutions is used to verify the order of accuracy of two finite‐volume Euler and Navier–Stokes codes. The Premo code employs a node‐centred approach using unstructured meshes, while the Wind code employs a similar scheme on structured meshes. Both codes use Roe's upwind method with MUSCL extrapolation for the convective terms and central differences for the diffusion terms, thus yielding a numerical scheme that is formally second‐order accurate. The method of manufactured solutions is employed to generate exact solutions to the governing Euler and Navier–Stokes equations in two dimensions along with additional source terms. These exact solutions are then used to accurately evaluate the discretization error in the numerical solutions. Through global discretization error analyses, the spatial order of accuracy is observed to be second order for both codes, thus giving a high degree of confidence that the two codes are free from coding mistakes in the options exercised. Examples of coding mistakes discovered using the method are also given. Copyright © 2004 John Wiley & Sons, Ltd.     

A robust low‐dissipation AUSM‐family scheme for numerical shock stability on unstructured grids

The simple low‐dissipation advection upwind splitting method (SLAU) scheme is a parameter‐free, low‐dissipation upwind scheme that has been applied in a wide range of aerodynamic numerical simulations. In spite of its successful applications, the SLAU scheme could be showing shock instabilities on unstructured grids, as many other contact resolved upwind schemes. Therefore, a hybrid upwind flux scheme is devised for improving the shock stability of SLAU scheme, without compromising on accuracy and low Mach number performance. Numerical flux function of the hybrid scheme is written in a general form, in which only the scalar dissipation term is different from that of the SLAU scheme. The hybrid dissipation term is defined by using a differentiable multidimensional‐shock‐detection pressure weight function, and the dissipation term of SLAU scheme is combined with that of the Van Leer scheme. Furthermore, the hybrid dissipation term is only applied for the solution of momentum fluxes in numerical flux function. Based on the numerical test results, the hybrid scheme is deemed to be a successful improvement on the shock stability of SLAU scheme, without compromising on the efficiency and accuracy. Copyright © 2016 John Wiley & Sons, Ltd.     

A stable gradient reconstruction for the MUSCL schemes applied to systems of conservation laws

The numerical simulations of compressible flows need more and more accuracy. Several approaches can be used to increase the order of the scheme. One of the most popular has been introduced by [11]: the so-called MUSCL scheme. In the present work, we focus our attention on the robustness of this method. We propose a relevant CFL restriction and a new gradient reconstruction strategy to enforce stability of the method. The novelty stays in the fact that non conservation argument is involved in the gradient reconstruction. Numerical results are performed using this new variant of the MUSCL scheme.     

极坐标系下Fourier-Legendre谱元方法与有限差分法数值扩散的比较

提出一种Fourier-Legendre谱元方法用于求解极坐标系下的Navier-Stokes方程,其中极点所在单元的径向采用Gauss-Radau积分点,避免了r=0处的1/r坐标奇异性。时间离散采用时间分裂法,引入数值同位素模型跟踪同位素的输运过程验证数值模拟的精度,分别利用谱元法和有限差分法的迎风差分格式求解匀速和加速坩埚旋转流动中的同位素方程。计算结果表明,有限差分法中的一阶迎风差分格式存在严重的数值假扩散,二阶迎风差分格式的数值结果较精确,增加节点可以有效地缓解数值扩散。然而,谱元法具有以较少节点得到高精度解的优势。     

Solution of the 2‐D shallow water equations with source terms in surface elevation splitting form

A vertex‐centred finite‐volume/finite‐element method (FV/FEM) is developed for solving 2‐D shallow water equations (SWEs) with source terms written in a surface elevation splitting form, which balances the flux gradients and source terms. The method is implemented on unstructured grids and the numerical scheme is based on a second‐order MUSCL‐like upwind Godunov FV discretization for inviscid fluxes and a classical Galerkin FE discretization for the viscous gradients and source terms. The main advantages are: (1) the discretization of SWE written in surface elevation splitting form satisfies the exact conservation property (??‐Property) naturally; (2) the simple centred‐type discretization can be used for the source terms; (3) the method is suitable for both steady and unsteady shallow water problems; and (4) complex topography can be handled based on unstructured grids. The accuracy of the method was verified for both steady and unsteady problems, including discontinuous cases. The results indicate that the new method is accurate, simple, and robust. Copyright © 2007 John Wiley & Sons, Ltd.     

Vortex formation mechanisms in the wake behind a sphere for 200 < Re < 380

Direct numerical simulation and visualization of three-dimensional separated flows of a homogeneous incompressible viscous fluid are used to comprehensively describe different mechanisms of vortex formation behind a sphere at moderate Reynolds numbers (200 ≤ Re ≤ 380). For 200 < Re ≤ 270 a steady-state rectilinear double-filament wake is formed, while for Re > 270 it is a chain of vortex loops. The three unsteady periodic flow patterns corresponding to the 270 < Re ≤ 290, 290 < Re ≤ 320, and 320 < Re ≤ 380 ranges are characterized by different vortex formation mechanisms. Direct numerical simulation is based on the Meranzh (SMIF) method of splitting in physical factors with an explicit hybrid finite-difference scheme which possesses the following properties: secondorder approximation in the spatial variables, minimal scheme viscosity and dispersion, and monotonicity. Two different vortex identification techniques are used for visualizing the vortex structures within the wake.     

Higher-order Upwind Distribution-formula Scheme for Structured and Unstructured Adaptive Flow Solvers

The development of inviscid and viscous flow solvers for both structured and unstructured meshes is presented in this paper. The solution method is the distribution-formula scheme. This is an explicit, cell-vertex, finite volume method which is essentially second-order accurate in both space and time. The Euler and Navier-Stokes equations are integrated over each finite volume cell to determine the change in flow properties (e.g. density) for the cell. Distribution formulas are then used to distribute such changes to the surrounding vertices. Increments in each vertex (which is a calculation point) thus consist of contributions from the surrounding cells. The original discretization technique involves central differencing and is simple, robust and computationally efficient. In this work, starting with inviscid flow simulations using the original scheme on structured grids, improvements are subsequently made to the scheme by replacing the central differencing portion with MUSCL type higher-order upwind differencing. Numerical investigations with the improved scheme are performed using inviscid flow simulations on structured grids. Upon establishing improved accuracy, stability and excellent shock capturing properties, further extension to viscous flow computations on unstructured adaptive meshes is implemented. Results are presented for laminar, viscous flow over a NACA 0012 airfoil.     

Assessment of TVD schemes for inviscid and turbulent flow computation

A systematic study has been conducted to assess the performance of the TVD schemes for practical flow computation. The viewpoint adopted here is to treat the TVD schemes as a combination of the standard central difference scheme with numerical dissipation terms. The controlled amount of numerical dissipation modifies the computed fluxes to ensure that the solution is oscillation-free. Four variants of TVD schemes, two with upwind dissipation terms and two with symmetric dissipation terms, have been studied and compared with the conventional Beam-Warming scheme for inviscid and turbulent axisymmetric flow computations. The results obtained show that all four variants can accurately resolve the shock and flow profiles with fewer grid points than the Beam-Warming scheme. The convergence rates of the TVD schemes are also substantially superior to that of the Beam-Warming scheme. The combination of high accuracy, good robustness and improved computational efficiency offered by the TVD schemes makes them attractive for computing high-speed flow with shocks. In terms of the relative performances it is found that the symmetric schemes converge slightly faster but that the upwind schemes are less sensitive to the number of grid points being employed.     

全机绕流Euler方程多重网格分区计算方法

全机三维复杂形状绕流数值求解只能采用分区求解的方法,本文采用可压缩Euler方程有限体积方法以及多重网格分区方法对流场进行分区计算。数值方法采用改进的van Leer迎风型矢通量分裂格式和MUSCL方法,基于有限体积方法和迎风型矢通量分裂方法,建立一套处理子区域内分界面的耦合条件。各个子区域之间采用显式耦合条件,区域内部采用隐式格式和局部时间步长等,以加快收敛速度。计算结果飞机表面压力分布等气动力特性与实验值进行了比较,二者基本吻合。计算结果表明采用分析“V”型多重网格方法,能提高计算效率,加快收敛速度达到接近一个量级。根据全机数值计算结果和可视化结果讨论了流场背风区域旋涡的形成过程。     

同位网格摄动有限体积格式求解浮力驱动方腔流

利用对流扩散方程的摄动有限体积格式,在Rayleigh数从10$^{3}$ 到10$^{8}$的范围内对浮力驱动方腔流动问题作了数值模拟. 对流扩散方程的摄动 有限体积格式具有一阶迎风格式的简洁形式,使用相同的基点,重构近似精度高,特别是两 相邻控制体中心到公共界面的距离相等或不相等,PFV格式公式相同等优点. 在数值模拟中, 无论均匀网格还是非均匀网格均获得与DSC方法、自适应有限元法、多重网格法等Benchmark 解相符较好的数值结果,证明UPFV格式对高Rayleigh数对流传热问题的适用性和有效性.     

Developing shock-capturing difference methods

A new shock-capturing method is proposed which is based on upwind schemes and flux-vector splittings. Firstly, original upwind schemes are projected along characteristic directions. Secondly, the amplitudes of the characteristic decompositions are carefully controlled by limiters to prevent non-physical oscillations. Lastly, the schemes are converted into conservative forms, and the oscillation-free shock-capturing schemes are acquired. Two explicit upwind schemes (2nd-order and 3rd-order) and three compact upwind schemes (3rd-order, 5th-order and 7th-order) are modified by the method for hyperbolic systems and the modified schemes are checked on several one-dimensional and two-dimensional test cases. Some numerical solutions of the schemes are compared with those of a WENO scheme and a MP scheme as well as a compact-WENO scheme. The results show that the method with high order accuracy and high resolutions can capture shock waves smoothly.     

不可压N-S方程高效算法及二维槽道湍流分析

构造了基于非等距网格的迎风紧致格式,并将其与三阶精度的Adams半隐方法相结合,构造了求解不可压N－S方程高效算法。该算法利用基于交错网格的离散形式的压力Poisson方程求解压力项,解决了边界处的残余散度问题;同时还利用快速Fourier变换将方程的隐式部分解耦,离散后的代数方程组利用追赶法求解,大大减少了计算量。通过对二维槽道流动的数值模拟,证实了所构造的数值方法具有精度高,稳定性好,能抑制混淆误差等优点,同时具有很高的计算效率,是进行壁湍流直接数值模拟的有效方法。在数值模拟的基础上对二维槽道流动进行了分析,得到了Reynolds数从6000到15000的二维流动饱和态解（所谓“二维槽道湍流”）;定性及定量结果均与他人的数值计算结果吻合十分理想。对流场进行了分析,指出了“二维湍流”与三维湍流统计特性的区别。     

EFFICIENT COMPUTATIONS USING UPWIND BIASED SCHEMES

A new multiblock unfactored implicit upwind scheme for inviscid two-dimensional flow calculations is presented. Spatial discretization is carried out by means of an upwind first-order method; an original extension to higher accuracy is also presented. The integration algorithm is constructed in a ‘δ’ form that provides a direct derivation of the scheme and leads to an efficient computational method. Fast solutions of the linear systems arising at each time step are obtained by means of the bi-conjugate gradient stabilized technique. The computational results for super/hypersonic steady state flows illustrate the efficiency and accuracy of the algorithm.     

对流扩散方程的迎风变换及相应有限差分方法

本文提出所谓迎风变换,将对流扩散方程分解为对流迎风函数和扩散方程,并构造相应的有限差分格式。对流迎风函数以简明的指数解析形式反映对流扩散现象的迎风效应,原则上消除了源于不对称对流算子的困难,能够便利对流扩散方程的数值求解。有限差分格式具有二阶精度和无条件稳定性,算例表明其准确性、收敛速度及对边界层效应的适应能力均明显优于中心差分格式和迎风差分格式。