A refined r‐factor algorithm for TVD schemes on arbitrary unstructured meshes

A refined r‐factor algorithm for implementing total variation diminishing (TVD) schemes on arbitrary unstructured meshes, referred to henceforth as a face‐perpendicular far‐upwind interpolation scheme for arbitrary meshes (FFISAM), is proposed based on an extensive review of the existing r‐factor algorithms available in the literature. The design principles, as well as the respective advantages and disadvantages, of the existing algorithms are first systematically analyzed before presenting the FFISAM. The FFISAM is designed to combine the merits of various existing r‐factor algorithms. The performance of the FFISAM, implemented in 10 classical TVD schemes, is evaluated against four two‐dimensional pure‐advection benchmark test cases where analytical solutions are available. The numerical results clearly show that the FFISAM leads to a better overall performance than the existing algorithms in terms of accuracy and convergence on arbitrary unstructured meshes for the 10 classical TVD schemes. Copyright © 2015 John Wiley & Sons, Ltd.     

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.     

Edge‐based reconstruction schemes for unstructured tetrahedral meshes

In this paper, we consider edge‐based reconstruction (EBR) schemes for solving the Euler equations on unstructured tetrahedral meshes. These schemes are based on a high‐accuracy quasi‐1D reconstruction of variables on an extended stencil along the edge‐based direction. For an arbitrary tetrahedral mesh, the EBR schemes provide higher accuracy in comparison with most second‐order schemes at rather low computational costs. The EBR schemes are built in the framework of vertex‐centered formulation for the point‐wise values of variables. Here, we prove the high accuracy of EBR schemes for uniform grid‐like meshes, introduce an economical implementation of quasi‐one‐dimensional reconstruction and the resulting new scheme of EBR family, estimate the computational costs, and give new verification results. Copyright © 2015 John Wiley & Sons, Ltd.     

Higher‐order bounded differencing schemes for compressible and incompressible flows

In recent years, three higher‐order (HO) bounded differencing schemes, namely AVLSMART, CUBISTA and HOAB that were derived by adopting the normalized variable formulation (NVF), have been proposed. In this paper, a comparative study is performed on these schemes to assess their numerical accuracy, computational cost as well as iterative convergence property. All the schemes are formulated on the basis of a new dual‐formulation in order to facilitate their implementations on unstructured meshes. Based on the proposed dual‐formulation, the net effective blending factor (NEBF) of a high‐resolution (HR) scheme can now be measured and its relevance on the accuracy and computational cost of a HR scheme is revealed on three test problems: (1) advection of a scalar step‐profile; (2) 2D transonic flow past a circular arc bump; and (3) 3D lid‐driven incompressible cavity flow. Both density‐based and pressure‐based methods are used for the computations of compressible and incompressible flow, respectively. Computed results show that all the schemes produce solutions which are nearly as accurate as the third‐order QUICK scheme; however, without the unphysical oscillations which are commonly inherited from the HO linear differencing scheme. Generally, it is shown that at higher value of NEBF, a HR scheme can attain better accuracy at the expense of computational cost. Copyright © 2006 John Wiley & Sons, Ltd.     

High resolution NVD differencing scheme for arbitrarily unstructured meshes

The issue of boundedness in the discretisation of the convection term of transport equations has been widely discussed. A large number of local adjustment practices has been proposed, including the well‐known total variation diminishing (TVD) and normalised variable diagram (NVD) families of differencing schemes. All of these use some sort of an ‘unboundedness indicator’ in order to determine the parts of the domain where intervention in the discretisation practice is needed. These, however, all use the ‘far upwind’ value for each face under consideration, which is not appropriate for unstructured meshes. This paper proposes a modification of the NVD criterion that localises it and thus makes it applicable irrespective of the mesh structure, facilitating the implementation of ‘standard’ bounded differencing schemes on unstructured meshes. Based on this strategy, a new bounded version of central differencing constructed on the compact computational molecule is proposed and its performance is compared with other popular differencing schemes on several model problems. Copyright © 1999 John Wiley & Sons, Ltd.     

A nonlocal convective flux limiter for upwind‐biased finite volume simulations

Finite volume methods on structured and unstructured meshes commonly use second‐order, upwind‐biased linear reconstruction schemes to approximate the convective terms. Limiters are employed to reduce the inherent variable overshoot and undershoot of these schemes in discontinuous regions; however, they also can significantly increase the numerical dissipation of a solution in smooth regions. This paper presents a novel nonlocal, nonmonotonic (NLNM) limiter developed by enforcing cell minima and maxima on dependent variable values projected to cell faces. The minimum and maximum values for a cell are determined primarily through recursive reference to the minimum and maximum values of its upwind neighbors. The new limiter has been implemented into an existing flow solver. Various test cases are presented, which highlight the ability of the NLNM limiter to eliminate nonphysical oscillations while maintaining negligible levels of limiter‐induced dissipation into the solution. Results from the new limiter are compared with those from other limited and unlimited second‐order upwind and first‐order upwind schemes. For the cases examined in the study, the NLNM limiter was found to improve accuracy without significantly decreasing overall simulation efficiency and robustness.Copyright © 2011 John Wiley & Sons, Ltd.     

An analytical interface reconstruction algorithm in the PLIC‐VOF method for 2D polygonal unstructured meshes

Piecewise linear interface calculation (PLIC) schemes have been extensively employed in the volume‐of‐fluid (VOF) method for interface capturing in numerical simulations of multiphase flows. Polygonal unstructured meshes are often adopted because of their geometric flexibility and superiority in gradient calculation. An analytical interface reconstruction algorithm in the PLIC‐VOF method for arbitrary convex polygonal cells has been proposed in this study. The line interface at a given orientation within a polygonal cell is located by an analytical technique. It has been tested successfully for four different geometric shapes that are common in polygonal meshes. The computational efficiency of the present algorithm has been compared with several published schemes in the literature. The proposed algorithm has been shown to yield higher accuracy with reduction in computational complexity. A numerical simulation of a dam‐breaking problem has been performed using the proposed analytical PLIC technique on polygonal meshes. The results are in good agreement with experimental data available in the literature, which serves as a demonstration of its performance in a real multiphase flow.     

Numerical analysis of conservative unstructured discretisations for low Mach flows

Unstructured meshes allow easily representing complex geometries and to refine in regions of interest without adding control volumes in unnecessary regions. However, numerical schemes used on unstructured grids have to be properly defined in order to minimise numerical errors. An assessment of a low Mach algorithm for laminar and turbulent flows on unstructured meshes using collocated and staggered formulations is presented. For staggered formulations using cell‐centred velocity reconstructions, the standard first‐order method is shown to be inaccurate in low Mach flows on unstructured grids. A recently proposed least squares procedure for incompressible flows is extended to the low Mach regime and shown to significantly improve the behaviour of the algorithm. Regarding collocated discretisations, the odd–even pressure decoupling is handled through a kinetic energy conserving flux interpolation scheme. This approach is shown to efficiently handle variable‐density flows. Besides, different face interpolations schemes for unstructured meshes are analysed. A kinetic energy‐preserving scheme is applied to the momentum equations, namely, the symmetry‐preserving scheme. Furthermore, a new approach to define the far‐neighbouring nodes of the quadratic upstream interpolation for convective kinematics scheme is presented and analysed. The method is suitable for both structured and unstructured grids, either uniform or not. The proposed algorithm and the spatial schemes are assessed against a function reconstruction, a differentially heated cavity and a turbulent self‐igniting diffusion flame. It is shown that the proposed algorithm accurately represents unsteady variable‐density flows. Furthermore, the quadratic upstream interpolation for convective kinematics scheme shows close to second‐order behaviour on unstructured meshes, and the symmetry‐preserving is reliably used in all computations. Copyright © 2016 John Wiley & Sons, Ltd.     

An efficient high order direct ALE ADER finite volume scheme with a posteriori limiting for hydrodynamics and magnetohydrodynamics

In this paper, we present a new family of direct arbitrary–Lagrangian–Eulerian (ALE) finite volume schemes for the solution of hyperbolic balance laws on unstructured meshes in multiple space dimensions. The scheme is designed to be high‐order accurate both in space and time, and the mesh motion, which provides the new mesh configuration at the next time step, is taken into account in the final finite volume scheme that is based directly on a space‐time conservation formulation of the governing PDE system. To improve the computational efficiency of the algorithm, high order of accuracy in space is achieved using the a posteriori MOOD limiting strategy that allows the reconstruction procedure to be carried out with only one reconstruction stencil for any order of accuracy. We rely on an element‐local space‐time Galerkin finite element predictor on moving curved meshes to obtain a high‐order accurate one‐step time discretization, while the mesh velocity is computed by means of a suitable nodal solver algorithm that might also be supplemented with a local rezoning procedure to improve the mesh quality. Next, the old mesh configuration at time level tⁿ is connected to the new one at t^n + 1 by straight edges, hence providing unstructured space‐time control volumes, on the boundary of which the numerical flux has to be integrated. Here, we adopt a quadrature‐free integration, in which the space‐time boundaries of the control volumes are split into simplex sub‐elements that yield constant space‐time normal vectors and Jacobian matrices. In this way, the integrals over the simplex sub‐elements can be evaluated once and for all analytically during a preprocessing step. We apply the new high‐order direct ALE algorithm to the Euler equations of compressible gas dynamics (also referred to as hydrodynamics equations) as well as to the magnetohydrodynamics equations and we solve a set of classical test problems in two and three space dimensions. Numerical convergence rates are provided up to fifth order of accuracy in 2D and 3D for both hyperbolic systems considered in this paper. Finally, the efficiency of the new method is measured and carefully compared against the original formulation of the algorithm that makes use of a WENO reconstruction technique and Gaussian quadrature formulae for the flux integration: depending on the test problem, the new class of very efficient direct ALE schemes proposed in this paper can run up to ≈12 times faster in the 3D case. Copyright © 2016 John Wiley & Sons, Ltd.     

Multi‐material incompressible flow simulation using the moment‐of‐fluid method

This paper compares the numerical performance of the moment‐of‐fluid (MOF) interface reconstruction technique with Youngs, LVIRA, power diagram (PD), and Swartz interface reconstruction techniques in the context of a volume‐of‐fluid (VOF) based finite element projection method for the numerical simulation of variable‐density incompressible viscous flows. In pure advection tests with multiple materials MOF shows dramatic improvements in accuracy compared with the other methods. In incompressible flows where density differences determine the flow evolution, all the methods perform similarly for two material flows on structured grids. On unstructured grids, the second‐order MOF, LVIRA, and Swartz methods perform similarly and show improvement over the first‐order Youngs' and PD methods. For flow simulations with more than two materials, MOF shows increased accuracy in interface positions on coarse meshes. In most cases, the convergence and accuracy of the computed flow solution was not strongly affected by interface reconstruction method. Published in 2009 by John Wiley & Sons, Ltd.     

A high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows

Adaptive mesh techniques are used widely in the numerical simulations of fluid flows, and the simulation results with high accuracies are obtained by appropriate mesh adaptations. However, gas–liquid two‐phase flows are still difficult to be simulated on adaptive meshes, especially on unstructured adaptive meshes, because the physical phenomena near gas–liquid interfaces are highly complicated and in general, not modeled appropriately on adaptive meshes. In this paper, a high‐precision unstructured adaptive mesh technique for gas–liquid two‐phase flows is developed and verified/validated. In the unstructured adaptive mesh technique, the PLIC algorithm is employed to simulate interfacial dynamic behaviors and, therefore, the reconstruction method for the interfaces in refined cells is developed, which satisfies the gas and liquid volume conservations and geometrical conservations of interfaces. In addition, the physics‐based consideration is performed on the momentum calculations near interfaces, and the calculation method with gas and liquid momentum conservations is developed. For verification, the slotted‐disk revolution problem is solved. As a result, the unstructured adaptive mesh technique succeeds in reproducing the slotted‐disk shape accurately and well maintaining the shape after one full‐revolution. The dam‐break problem is also simulated and the momentum conservative calculation method succeeds in providing physically appropriate results, which show good agreements with experimental data. Therefore, it is confirmed that the developed unstructured adaptive mesh technique is very efficient to simulate gas–liquid two‐phase flows accurately. Copyright © 2010 John Wiley & Sons, Ltd.     

Discretization of transport equations on 2D Cartesian unstructured grids using data from remote cells for the convection terms

This paper presents a new finite volume discretization methodology for the solution of transport equations on locally refined or unstructured Cartesian meshes. The implementation of the cell‐face values of the dependent variables enables the employment of data from remote cells and thus the use of higher‐order differencing schemes. It also results in simple and flux‐conservative multiple‐scale stencils for the discretization of the governing equations. The latter are finally cast into a generalized form that does not depend on the local mesh structure. The performance of the numerical model is demonstrated on some classical 2D problems using various gridding techniques and a bounded second‐order upwind scheme. A stable and efficient behaviour of the algorithm is observed in all test cases. The results indicate that the combination in the present model of both local grid refinement and second‐order discretization can produce substantially more accurate solutions than each of the above techniques alone, for the same computational effort. The method is also applicable to turbulent flows and can be easily extended to three‐dimensions. Copyright © 2003 John Wiley & Sons, Ltd.     

Convergence issues in using high‐resolution schemes and lower–upper symmetric Gauss–Seidel method for steady shock‐induced combustion problems

This paper reports numerical convergence study for simulations of steady shock‐induced combustion problems with high‐resolution shock‐capturing schemes. Five typical schemes are used: the Roe flux‐based monotone upstream‐centered scheme for conservation laws (MUSCL) and weighted essentially non‐oscillatory (WENO) schemes, the Lax–Friedrichs splitting‐based non‐oscillatory no‐free parameter dissipative (NND) and WENO schemes, and the Harten–Yee upwind total variation diminishing (TVD) scheme. These schemes are implemented with the finite volume discretization on structured quadrilateral meshes in dimension‐by‐dimension way and the lower–upper symmetric Gauss–Seidel (LU–SGS) relaxation method for solving the axisymmetric multispecies reactive Navier–Stokes equations. Comparison of iterative convergence between different schemes has been made using supersonic combustion flows around a spherical projectile with Mach numbers M = 3.55 and 6.46 and a ram accelerator with M = 6.7. These test cases were regarded as steady combustion problems in literature. Calculations on gradually refined meshes show that the second‐order NND, MUSCL, and TVD schemes can converge well to steady states from coarse through fine meshes for M = 3.55 case in which shock and combustion fronts are separate, whereas the (nominally) fifth‐order WENO schemes can only converge to some residual level. More interestingly, the numerical results show that all the schemes do not converge to steady‐state solutions for M = 6.46 in the spherical projectile and M = 6.7 in the ram accelerator cases on fine meshes although they all converge on coarser meshes or on fine meshes without chemical reactions. The result is based on the particular preconditioner of LU–SGS scheme. Possible reasons for the nonconvergence in reactive flow simulation are discussed.Copyright © 2012 John Wiley & Sons, Ltd.     

A novel multidimensional solution reconstruction and edge‐based limiting procedure for unstructured cell‐centered finite volumes with application to shallow water dynamics

On unstructured meshes, the cell‐centered finite volume (CCFV) formulation, where the finite control volumes are the mesh elements themselves, is probably the most used formulation for numerically solving the two‐dimensional nonlinear shallow water equations and hyperbolic conservation laws in general. Within this CCFV framework, second‐order spatial accuracy is achieved with a Monotone Upstream‐centered Schemes for Conservation Laws‐type (MUSCL) linear reconstruction technique, where a novel edge‐based multidimensional limiting procedure is derived for the control of the total variation of the reconstructed field. To this end, a relatively simple, but very effective modification to a reconstruction procedure for CCFV schemes, is introduced, which takes into account geometrical characteristics of computational triangular meshes. The proposed strategy is shown not to suffer from loss of accuracy on grids with poor connectivity. We apply this reconstruction in the development of a second‐order well‐balanced Godunov‐type scheme for the simulation of unsteady two‐dimensional flows over arbitrary topography with wetting and drying on triangular meshes. Although the proposed limited reconstruction is independent from the Riemann solver used, the well‐known approximate Riemann solver of Roe is utilized to compute the numerical fluxes, whereas the Green–Gauss divergence formulation for gradient computations is implemented. Two different stencils for the Green–Gauss gradient computations are implemented and critically tested, in conjunction with the proposed limiting strategy, on various grid types, for smooth and nonsmooth flow conditions. Copyright © 2012 John Wiley & Sons, Ltd.     

High‐order ENO and WENO schemes for unstructured grids

This work describes the implementation and analysis of high‐order accurate schemes applied to high‐speed flows on unstructured grids. The class of essentially non‐oscillatory schemes (ENO), that includes weighted ENO schemes (WENO), is discussed in the paper with regard to the implementation of third‐ and fourth‐order accurate methods. The entire reconstruction process of ENO and WENO schemes is described with emphasis on the stencil selection algorithms. The stencils can be composed by control volumes with any number of edges, e.g. triangles, quadrilaterals and hybrid meshes. In the paper, ENO and WENO schemes are implemented for the solution of the dimensionless, 2‐D Euler equations in a cell centred finite volume context. High‐order flux integration is achieved using Gaussian quadratures. An approximate Riemann solver is used to evaluate the fluxes on the interfaces of the control volumes and a TVD Runge–Kutta scheme provides the time integration of the equations. Such a coupling of all these numerical tools, together with the high‐order interpolation of primitive variables provided by ENO and WENO schemes, leads to the desired order of accuracy expected in the solutions. An adaptive mesh refinement technique provides better resolution in regions with strong flowfield gradients. Results for high‐speed flow simulations are presented with the objective of assessing the implemented capability. Copyright © 2007 John Wiley & Sons, Ltd.     

多维迎风方法在非结构/混合网格热流计算中的应用研究

非结构/混合网格具有极强的几何灵活性,在复杂外形飞行器的气动力特性数值模拟中已得到广泛应用,但目前还难以准确地预测气动热环境。本文从非结构/混合网格热流计算的三个需求出发,选取了多维迎风方法,并与其他方法进行了对比研究。以二维圆柱高超声速绕流这一Benchmark典型问题为例,对比研究了多维迎风方法和几种广泛使用的无粘通量格式(Roe格式、Van Leer格式和AUSMDV格式)对混合网格热流计算精度的影响。结果表明,多维迎风方法在热流计算精度、鲁棒性以及收敛性方面表现良好。最后,将多维迎风方法应用于常规混合网格上的圆柱和钝双锥绕流问题,均得到了较好的热流计算结果,为非结构/混合网格热流计算在复杂高超飞行器中的应用奠定了基础。     

Analytical interface reconstruction algorithms in the PLIC-VOF method for 3D polyhedral unstructured meshes

Piecewise linear interface calculation (PLIC) schemes have been extensively employed in the volume-of-fluid (VOF) method for interface capturing in numerical simulations of multiphase flows. Polyhedral unstructured meshes are often adopted due to their geometric flexibility and superiority in gradient calculation. Four analytical interface reconstruction algorithms in the PLIC-VOF method for arbitrary convex polyhedral cells have been proposed in this study. The plane interface at a given orientation within a polyhedral cell is located by four different analytical techniques. They have been tested successfully for six different geometric shapes that are common in polyhedral meshes. The computational efficiencies of the algorithms have been compared with two other published schemes in the literature. The proposed algorithms have been shown to yield smaller truncation errors with reduction in computational complexity. A numerical simulation of a 3D dam-breaking problem has been successfully performed using the proposed interface reconstruction scheme on a polyhedral mesh. The percentage of the overall computational time consumed has been assessed to justify its optimization in a real multiphase flow simulation.     

Basic function scheme of polynomial type

A new numerical method named as basic function method is proposed. It can directly discretize differential operators on unstructured grids. By expanding the basic function to approach the exact function, the central and upwind schemes of derivative are constructed. By using the second-order polynomial as a basic function and applying the flux splitting method and the combination of central and upwind schemes to suppress non-physical fluctuation near shock waves, a second-order basic function scheme of polynomial type is proposed to solve inviscid compressible flows numerically. Numerical results of typical examples for two-dimensional inviscid compressible transonic and supersonic steady flows indicate that the new scheme has high accuracy and high resolution for shock waves. Combined with the adaptive remeshing technique, satisfactory results can be obtained.     

A second-order upwind finite-volume method for the Euler solution on unstructured triangular meshes

A scheme for the numerical solution of the two-dimensional (2D) Euler equations on unstructured triangular meshes has been developed. The basic first-order scheme is a cell-centred upwind finite-volume scheme utilizing Roe's approximate Riemann solver. To obtain second-order accuracy, a new gradient based on the weighted average of Barth and Jespersen's three-point support gradient model is used to reconstruct the cell interface values. Characteristic variables in the direction of local pressure gradient are used in the limiter to minimize the numerical oscillation around solution discontinuities. An Approximate LU (ALU) factorization scheme originally developed for structured grid methods is adopted for implicit time integration and shows good convergence characterisitics in the test. To eliminate the data dependency which prohibits vectorization in the inversion process, a black-gray-white colouring and numbering technique on unstructured triangular meshes is developed for the ALU factorization scheme. This results in a high degree of vectorization of the final code. Numerical experiments on transonic Ringleb flow, transonic channel flow with circular bump, supersonic shock reflection flow and subsonic flow over multielement aerofoils are calculated to validate the methodology.     

A convergent and universally bounded interpolation scheme for the treatment of advection

A high resolution scheme with improved iterative convergence properties was devised by incorporating total‐variation diminishing constraints, appropriate for unsteady problems, into an implicit time‐marching method used for steady flow problems. The new scheme, referred to as Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection (CUBISTA), has similar accuracy to the well‐known SMART scheme, both being formally third‐order accurate on uniform meshes for smooth flows. Three demonstration problems are considered: (1) advection of three scalar profiles, a step, a sine‐squared, and a semi‐ellipse; (2) Newtonian flow over a backward‐facing step; and (3) viscoelastic flow through a planar contraction and around a cylinder. For the case of the viscoelastic flows, in which the high resolution schemes are also used to represent the advective terms in the constitutive equation, it is shown that only the new scheme is able to provide a converged solution to the prescribed tolerance. Copyright © 2003 John Wiley & Sons, Ltd.