THE INFLUENCE OF GLOBAL-DIRECTION STENCIL ON GRADIENT AND HIGH-ORDER DERIVATIVES RECONSTRUCTION OF UNSTRUCTURED FINITE VOLUME METHODS 1)

doi:10.6052/0459-1879-20-093

Abstract

Abstract:

The accuracy of unstructured finite volume methods is greatly influenced by different stencils. In previous work, based on the existing problems of local-direction stencil, we explored a more concise global-direction stencil selection method for the second-order unstructured finite volume solver, and stencil cells selected by this novel stencil selection method are always along the boundary normal and circumferential directions even on grids with high aspect ratio. As a result, the variation of flowfield is effectively captured, and flow anisotropy are well reflected. In addition, the novel method is topology-independent, since global directions are determined by the flowfield, while the local directions are strongly coupled with the grid. Therefore, the complex process of advancing front as well as local directions estimation are completely avoided in the novel stencil selection method, and the phenomenon that stencil cells deviate from the boundary normal vector is effectively eliminated on high-aspect-ratio triangular grids. What's more, a better computational accuracy and lower truncation errors on the second-order accurate finite volume solver are obtained by the employment of global-direction stencil. In order to further test the effectiveness of global-direction stencil on high-order unstructured finite volume methods, we will preliminarily utilize this stencil to test the effect of gradient and high-order derivatives reconstruction. After verification, computational errors of global-direction stencil are lower than that of local-direction stencil, and also lower than that of commonly used vertex-neighbor stencil on different grid types. Besides, errors of variable and derivatives at the Gauss point obtained by global-direction stencil are also the lowest among three methods we tested. Therefore, the global-direction stencil is well performed on gradient as well as high-order derivatives reconstruction, and it is feasible to extend this novel stencil selection method to high-order unstructured finite volume methods.

Key words: high-order unstructured finite volume methods, $k$-exact reconstruction algorithm, stencil selection method, global direction, local direction

CLC Number:

V211.3

Kong Lingfa, Dong Yidao, Liu Wei. THE INFLUENCE OF GLOBAL-DIRECTION STENCIL ON GRADIENT AND HIGH-ORDER DERIVATIVES RECONSTRUCTION OF UNSTRUCTURED FINITE VOLUME METHODS ¹⁾[J]. Chinese Journal of Theoretical and Applied Mechanics, 2020, 52(5): 1334-1349.

Figures/Tables 24

Fig.1 Unstructured finite volume discretization and flux construction

Fig.2 Face-neighbor and vertex-neighbor stencils

Fig.3 Possible combinations of local directions on triangular cell

Fig.4 Local directions of triangular cell

Fig.5 Local-direction stencil cells on triangular grids with different aspect ratios

Fig.6 Global-direction stencil cells on triangular grids with different aspect ratios

Fig.7 Schematic diagram of computational grids

Table 1 The distribution of background quadrilateral grids

Fig.8 $L_2$ norm of gradient errors on regular grids

Fig.9 $L_2$ norm of high-order derivative errors on regular grids

Table 2 $L_2$ errors of derivatives on the vfin grid

Fig.10 $L_2$ norm of gradient errors at Gauss point

Fig.11 $L_2$ norm of variable errors at Gauss point

Table 3 $L_2$ errors of variable and derivatives at Gauss points

Fig.12 $L_2$ norm of gradient errors on perturbed grids

Fig.13 $L_2$ norm of high-order derivative errors on perturbed grids

Fig.14 $L_2$ norm of gradient errors at Gauss point

Fig.15 $L_2$ norm of variable errors at Gauss point

Fig.16 $L_2$ norm of gradient errors on randomly-diag perturbed grids

Fig.17 $L_2$ norm of high-order derivative errors on randomly-diag perturbed grids

Table 4 $L_2$ errors of derivatives on the vfin grid

Fig.18 $L_2$ norm of gradient errors at Gauss point

Fig.19 $L_2$ norm of variable errors at Gauss point

Table 5 $L_2$ errors of variable and derivatives at Gauss points

References 39

[1]	Mavriplis DJ. Unstructured grid techniques. Annual Review of Fluid Mechanics, 1997,29:473-547 DOI URL
[2]	曾扬兵, 沈孟育, 王保国等. N-S方程在非结构网格下的求解. 力学学报, 1996,28(6):641-650
	( Zeng Yangbing, Shen Mengyu, Wang Baoguo, et al. Solution of N-S equations on unstructured grid. Chinese Journal of Theoretical and Applied Mechanics, 1996,28(6):641-650 (in Chinese))
[3]	王年华, 李明, 张来平. 非结构网格二阶有限体积法中黏性通量离散格式精度分析与改进. 力学学报, 2018,50(3):527-537
	( Wang Nianhua, Li Ming, Zhang Laiping. Accuracy analysis and improvement of viscous flux schemes in unstructured second-order finite-volume discretization. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(3):527-537 (in Chinese))
[4]	Dufresne Y, Moureau V, Lartigue G, et al. A massively parallel CFD/DEM approach for reactive gas-solid flows in complex geometries using unstructured meshes. Computers & Fluids, 2020,198:104402
[5]	王年华, 常兴华, 马戎等. HyperFlow软件非结构网格亚跨声速湍流模拟的验证与确认. 力学学报, 2019,51(3):813-825
	( Wang Nianhua, Chang Xinghua, Ma Rong, et al. Verification and validation of HyperFlow solver for subsonic and transonic turbulent flow simulations on unstructured/hybrid grids. Chinese Journal of Theoretical and Applied Mechanics, 2019,51(3):813-825 (in Chinese))
[6]	Park MA, Balan A, Anderson WK, et al. Verification of unstructured grid adaptation components// AIAA Scitech 2019 Forum, San Diego, 2019-1-7-11, 2019: 1723
[7]	邵帅, 李明, 王年华等. 基于非结构/混合网格模拟黏性流的高阶精度DDG/FV 混合方法. 力学学报, 2018,50(6):1470-1482
	( Shao Shuai, Li Ming, Wang Nianhua, et al. High-order DDG/FV hybrid method for viscous flow simulation based on unstructured / hybrid grid. Chinese Journal of Theoretical and Applied Mechanics, 2018,50(6):1470-1482 (in Chinese))
[8]	Diskin B, Thomas JL, Nielsen EJ, et al. Comparison of node-centered and cell-centered unstructured finite-volume discretizations: Viscous fluxes. AIAA Journal, 2010,48:1326-1338
[9]	Diskin B, Thomas JL. Comparison of node-centered and cell-centered unstructured finite-volume discretizations: Inviscid fluxes. AIAA Journal, 2011,49:836-854 DOI URL
[10]	Jameson A, Mavriplis DJ. Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh. AIAA Journal, 1986,24:611-618
[11]	Caraeni D, Hill DC. Unstructured-grid third-order finite volume discretization using a multistep quadratic data-reconstruction method. AIAA Journal, 2010,48:808-817
[12]	Schwoppe A, Diskin B. Accuracy of the cell-centered grid metric in the DLR TAU-Code. Notes on Numerical Fluid Mechanics and Multidisciplinary Design, 2013,121:429-437
[13]	贺立新, 张来平, 张涵信. 间断Galerkin有限元和有限体积. 力学学报, 2007,23(1):15-22
	( He Lixin, Zhang Laiping, Zhang Hanxin. Discontinuous Galerkin finite element and finite volume. Chinese Journal of Theoretical and Applied Mechanics, 2007,23(1):15-22 (in Chinese))
[14]	Mavriplis DJ. Multigrid solution of the two-dimensional Euler equations on unstructured triangular meshes. AIAA Journal, 1988,26:824-831
[15]	赵辉, 张耀冰, 陈江涛等. 非结构网格体心梯度求解方法的精度分析. 空气动力学学报, 2019,37(5):844-854
	( Zhao Hui, Zhang Yaobing, Chen Jiangtao, et al. Accuracy analysis of the method for solving the cell-centered gradient on unstructured grid. Acta Aerodynamica Sinica, 2019,37(5):844-854 (in Chinese))
[16]	Shima E, Kitamura K, Haga T. Green-gauss/weighted-least-squares hybrid gradient reconstruction for arbitrary polyhedra unstructured grids. AIAA Journal, 2013,51:2740-2747
[17]	王亚辉, 刘伟, 袁礼等. 求解二维Euler方程有限单元边插值的降维重构算法. 气体物理, 2019,3:34-41
	( Wang Yahui, Liu Wei, Yuan Li, et al. A dimension reduction reconstruction algorithm for solving two-dimensional Euler equations with finite element edge interpolation. Physics of Gases, 2019,3:34-41 (in Chinese))
[18]	Moukalled F, Mangani L, Darwish M. The Finite Volume Method in Computational Fluid Dynamics. Berlin: Springer, 2016: 46-49
[19]	张帆. 非结构网格有限体积法的空间离散算法研究. [博士论文]. 大连: 大连理工大学, 2017
	( Zhang Fan. Research on spatial discretization algorithm of unstructured mesh finite volume method. [PhD Thesis]. Dalian: Dalian University of Technology, 2017 (in Chinese))
[20]	王年华, 张来平, 赵钟等. 基于制造解的非结构二阶有限体积离散格式的精度测试与验证. 力学学报, 2017,49(3):627-637
	( Wang Nianhua, Zhang Laiping, Zhao Zhong, et al. Accuracy verification of unstructured second-order finite volume discretization schemes based on the method of manufactured solutions. Chinese Journal of Theoretical and Applied Mechanics, 2017,49(3):627-637 (in Chinese))
[21]	Sokolova I, Bastisya MG, Hajibeygi H. Multiscale finite volume method for finite-volume-based simulation of poroelasticity. Journal of Computational Physics, 2019,379:309-324.
[22]	White JA, Nishikawa H, Baurle RA. Weighted least-squares cell-average gradient construction methods for The VULCAN-CFD second-order accurate unstructured grid cell-centered finite-volume solver// AIAA Scitech 2019 Forum, San Diego, 2019-1-7-11, 2019: 0127
[23]	Nishikawa H, White JA. Face-and cell-averaged nodal-gradient approach to cell-centered finite-volume method on mixed grids// AIAA Scitech 2020 Forum, Orlando, 2020-1-6-10, 2020: 1787
[24]	Nishikawa H. Efficient gradient stencils for robust implicit finite-volume solver convergence on distorted grids. Journal of Computational Physics, 2019,386:486-501
[25]	Barth TJ, Jaespersen DC. The design and application of upwind schemes on unstructured meshes// 27th Aerospace Sciences Meeting, Nevada, 1989-1-9-12, 1989: 0366
[26]	Barth TJ. A 3-D upwind Euler solver for unstructured meshes// 10th Computational Fluid Dynamics Conference. Hawaii, 1991-1-24-27, 1991: 1548
[27]	Jalali A, Ollivier-Gooch CF. Higher-order unstructured finite volume RANS solution of turbulent compressible flows. Computers & Fluids, 2017,143:32-47
[28]	Ollivier-Gooch CF. Analysis of unstructured meshes from GMGW-1/HiLiftPW-3//2018 AIAA Aerospace Sciences Meeting, Florida. 2018-1-8-10, 2018: 0132
[29]	Jalali A, Ollivier-Gooch CF. Higher-order unstructured finite volume methods for turbulent aerodynamic flows// 22nd AIAA Computational Fluid Dynamics Conference, Dallas, 2015-1-22-26, 2015: 2284
[30]	Jalali A, Ollivier-Gooch CF. Higher-order least-squares reconstruction for turbulent aerodynamic flows// 11th World Congress on Computational Mechanics, Spain, 2014-7-20-25, 2014: 20-25
[31]	Zangeneh R, Ollivier-Gooch CF. Stability analysis and improvement of the solution reconstruction for cell-centered finite volume methods on unstructured meshes. Journal of Computational Physics, 2019,393:375-405
[32]	Ollivier-Gooch CF, Nejat A, Michalak K. Obtaining and verifying high-order unstructured finite volume solutions to the euler equations. AIAA Journal, 2009,47:2105-2120 DOI URL
[33]	Jalali A, Ollivier-Gooch CF. Higher-order finite volume solution reconstruction on highly anisotropic meshes// 21st AIAA Computational Fluid Dynamics Conference, California, 2013-6-24-27, 2013: 2565
[34]	Sozer E, Brehm C, Kiris CC. Gradient calculation methods on arbitrary polyhedral unstructured meshes for cell-centered cfd solvers// 52nd Aerospace Science Meeting, Washington DC, 2014-1-13-17, 2014: 1440
[35]	Xiong M, Deng XG, Gao X, et al. A novel stencil selection method for the gradient reconstruction on unstructured grid based on OpenFOAM. Computers & Fluids, 2018,172:426-442
[36]	Mavriplis DJ. An advancing front Delaunay triangulation algorithm designed for robustness. Journal of Computational Physics, 1995,117:90-101 DOI URL
[37]	Luo SH. A new mesh generation scheme for arbitrary planar domains. International Journal for Numerical Methods in Engineering, 1985,21:1403-1426 DOI URL
[38]	Li XS, Ren XD, Gu CW, et al. Shock-stable roe scheme combining entropy fix and rotated Riemann solver. AIAA Journal, 2020,58:779-786 DOI URL
[39]	Roe PL. Characteristic-based schemes for the Euler equations. Annual Review of Fluid Mechanics, 1986,18:337-365

THE INFLUENCE OF GLOBAL-DIRECTION STENCIL ON GRADIENT AND HIGH-ORDER DERIVATIVES RECONSTRUCTION OF UNSTRUCTURED FINITE VOLUME METHODS ¹⁾

RichHTML

PDF (PC)

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 24

References 39

Related Articles 15

Recommended Articles

Metrics

Comments

[1]	. PRELIMINARY INVESTIGATION ON UNSTRUCTURED MESH GENERATION TECHNIQUE BASED ON ADVANCING FRONT METHOD AND MACHINE LEARNING METHODS [J]. Chinese Journal of Theoretical and Applied Mechanics, 0, (): 0-0.
[2]	. NUMERICAL INVESTIGATION OF THE EVOLUTION OF TWO-DIMENSIONAL T-S WAVES ON AN ANISOTROPIC COMPLIANT WALL [J]. Chinese Journal of Theoretical and Applied Mechanics, 0, (): 0-0.
[3]	. The influence of the capsule on the supersonic rigid disk-gap-band parachute system [J]. Chinese Journal of Theoretical and Applied Mechanics, 0, (): 0-0.
[4]	. A FINITE VOLUME METHOD WITH WALSH BASIS FUNCTIONS TO CAPTURE DISCONTINUITY INSIDE GRID [J]. Chinese Journal of Theoretical and Applied Mechanics, 0, (): 0-0.
[5]	. The Influence of Global-Direction Stencil on Gradient and High-Order Derivatives of Unstructured Finite Volume Methods [J]. Chinese Journal of Theoretical and Applied Mechanics, 0, (): 0-0.
[6]	Luo Xin, Wu Songping. AN IMPROVED FIFTH-ORDER WENO-Z+ SCHEME ¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(6): 1927-1939.
[7]	Nianhua Wang, Xinghua Chang, Rong Ma, Laiping Zhang. VERIFICATION AND VALIDATION OF HYPERFLOW SOLVER FOR SUBSONIC AND TRANSONIC TURBULENT FLOW SIMULATIONS ON UNSTRUCTURED/HYBRID GRIDS¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(3): 813-825.
[8]	Qiao Chenliang, Xu Heyong, Ye Zhengyin. CIRCULATION CONTROL ON WIND TURBINE AIRFOIL WITH BLUNT TRAILING EDGE [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 135-145.
[9]	Liu Huixiang, He Guoyi, Wang Qi. NUMERICAL STUDY ON THE AERODYNAMIC PERFORMANCE OF THEFLEXIBLE AND CORRUGATED FOREWING OF DRAGONFLY IN GILDINGFLIGHT [J]. Chinese Journal of Theoretical and Applied Mechanics, 2019, 51(1): 94-102.
[10]	Shao Shuai, Li Ming, Wang Nianhua, Zhang Laiping. HIGH-ORDER DDG/FV HYBRID METHOD FOR VISCOUS FLOW SIMULATION ON UNSTRUCTURED/HYBRID GRIDS¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(6): 1470-1482.
[11]	Ye Youda, Zhang Hanxin, Jiang Qinxue, Zhang Xianfeng. SOME KEY PROBLEMS IN THE STUDY OF AERODYNAMIC CHARACTERISTICS OF NEAR-SPACE HYPERSONIC VEHICLES¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(6): 1292-1310.
[12]	Wan Yunbo, Ma Rong, Wang Nianhua, Zhang Laiping, Gui Yewei. ACCURATE AERO-HEATING PREDICTIONS BASED ON MUL-TI-DIMENSIONAL GRADIENT RECONSTRUCTION ON HYBRID UNSTRUCTURED GRIDS¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(5): 1003-1012.
[13]	Zhang Yang, Zou Jianfeng, Zheng Yao. AN IMPROVED GHOST-CELL IMMERSED BOUNDARY METHOD FOR SOLVING SUPERSONIC FLOW PROBLEMS [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(3): 538-552.
[14]	Wang Nianhua, Li Ming, Zhang Laiping. ACCURACY ANALYSIS AND IMPROVEMENT OF VISCOUS FLUX SCHEMES IN UNSTRUCTURED SECOND-ORDER FINITE-VOLUME DISCRETIZATION [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(3): 527-537.
[15]	Tong Fulin, Li Xin, Yu Changping, Li Xinliang. DIRECT NUMERICAL SIMULATION OF HYPERSONIC SHOCK WAVE AND TURBULENT BOUNDARY LAYER INTERACTIONS ¹⁾ [J]. Chinese Journal of Theoretical and Applied Mechanics, 2018, 50(2): 197-208.