Abstract
In this paper, we present a modified tangent of hyperbola for interface capturing (THINC) scheme for freestream and vortex preservation on the general curvilinear grids, in which the symmetric conservative metric method is employed to eliminate the geometrical errors in the discretization of Jacobian and the metrics. As the original THINC with a fixed jump steepness may lose accuracy and result in instability problem on non-uniform grids, a new algorithm is introduced to reduce the numerical dissipation, in which the jump steepness is scaled adaptively according to varying mesh intervals. Numerical tests show the new THINC scheme can hold freestream and vortex preservation and is capable of resolving discontinuities and small-scale smooth flow structures with less dissipation on general curvilinear grids, compared with the original THINC. By using the boundary variation diminishing (BVD) principle, the modified THINC is implemented in combination with a finite-difference weighted essentially non-oscillatory (WENO) scheme. Comprehensive numerical validations are performed to evaluate the performance of the improved THINC and WENO combined with the modified THINC in the framework of both the FDM and FVM, with twisting and even randomly spaced curvilinear meshes. Numerical results of the double Mach reflection also demonstrate the improved THINC can alleviate non-physical oscillations with less numerical dissipation.
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摘要
本文提出了一种改进的THINC数值方案, 用于在曲线网格下保持自由来流和等熵涡特性, 其中采用对称保守度量方法来消除坐标转换系数在数值离散中诱导的几何误差. 由于非等间距的网格会导致原本带有固定陡度参数的THINC方案的数值精度和计算稳定性降低. 因此, 本文引入了一种带有自适应陡度参数的THINC方案用于减少数值耗散, 其中陡度参数根据网格间隔的大小建立方程实现自适应缩放. 数值测试表明, 与原始THINC方案相比, 新的THINC方案能够有效地在一般结构化曲线网格下保持自由流和涡旋保存, 并且能够更加精准地拟合流场中的各类型间断和小尺度结构. 此外, 通过边界变差最小(BVD)原理, 将改进的THINC方案与加权本质非振荡方案(WENO)相结合, 兼顾了光滑解和间断解的数值精度. 本文在波浪形网格和随机网格下设置了一系列一维和二维数值算例, 分别测试了在有限差分和有限体积框架下改进的THINC方法与低耗散的WENO-THINC 综合方案的性能. 数值验证结果表明改进后的数值方案可以有效地减少非物理振荡和数值耗散, 相较于原始方案和同类型数值方案有较明显的优势.
References
T. H. Pulliam, and J. L. Steger, Implicit finite-difference simulations of three-dimensional compressible flow, AIAA J. 18, 159 (1980).
J. G. Trulio, and K. R. Trigger, Numerical solution of the one-dimensional hydrodynamic equations in an arbitrary time-dependent coordinate system (University of California Lawrence Radiation Laboratory Report UCLR-6522, 1961).
P. D. Thomas, and C. K. Lombard, Geometric conservation law and its application to flow computations on moving grids, AIAA J. 17, 1030 (1979).
M. R. Visbal, and D. V. Gaitonde, On the use of higher-order finite-difference schemes on curvilinear and deforming meshes, J. Comput. Phys. 181, 155 (2002).
X. Deng, M. Mao, G. Tu, H. Liu, and H. Zhang, Geometric conservation law and applications to high-order finite difference schemes with stationary grids, J. Comput. Phys. 230, 1100 (2011).
T. Nonomura, N. Iizuka, and K. Fujii, Freestream and vortex preservation properties of high-order WENO and WCNS on curvilinear grids, Comput. Fluids 39, 197 (2010).
T. Nonomura, D. Terakado, Y. Abe, and K. Fujii, A new technique for freestream preservation of finite-difference WENO on curvilinear grid, Comput. Fluids 107, 242 (2015).
Y. Zhu, Z. Sun, Y. Ren, Y. Hu, and S. Zhang, A numerical strategy for freestream preservation of the high order weighted essentially non-oscillatory schemes on stationary curvilinear grids, J. Sci. Comput. 72, 1021 (2017).
Y. Zhu, and X. Hu, Free-stream preserving linear-upwind and WENO schemes on curvilinear grids, J. Comput. Phys. 399, 108907 (2019), arXiv: 1811.06325.
W. F. Huang, Y. X. Ren, and X. Jiang, A simple algorithm to improve the performance of the WENO scheme on non-uniform grids, Acta Mech. Sin. 34, 37 (2018).
S. Rathan, and N. R. Gande, A modified fifth-order WENO scheme for hyperbolic conservation laws, Comput. Math. Appl. 75, 1531 (2018).
S. Rathan, and N. R. Gande, Improved weighted ENO scheme based on parameters involved in nonlinear weights, Appl. Math. Comput. 331, 120(2018).
N. R. Gande, and A. A. Bhise, Modified third and fifth order WENO schemes for inviscid compressible flows, Numer. Algor. 88, 249 (2021).
X. Yang, C. Liu, D. Wan, and C. Hu, Numerical study of the shock wave and pressure induced by single bubble collapse near planar solid wall, Phys. Fluids 33, 073311 (2021).
Y. Wang, and G. Dong, A numerical study of shock-interface interaction and prediction of the mixing zone growth in inhomogeneous medium, Acta Mech. Sin. 38, 122163 (2022).
S. Liu, X. Yuan, Z. Liu, Q. Yang, G. Tu, X. Chen, Y. Gui, and J. Chen, Design and transition characteristics of a standard model for hypersonic boundary layer transition research, Acta Mech. Sin. 37, 1637 (2021).
F. Xiao, Y. Honma, and T. Kono, A simple algebraic interface capturing scheme using hyperbolic tangent function, Int. J. Numer. Meth. Fluids 48, 1023 (2005).
B. Xie, S. Li, and F. Xiao, An efficient and accurate algebraic interface capturing method for unstructured grids in 2 and 3 dimensions: The THINC method with quadratic surface representation, Int. J. Numer. Meth. Fluids 76, 1025 (2014).
C. Liu, and C. Hu, Adaptive THINC-GFM for compressible multimedium flows, J. Comput. Phys. 342, 43 (2017).
Z. He, Y. Ruan, Y. Yu, B. Tian, and F. Xiao, Self-adjusting steepness-based schemes that preserve discontinuous structures in compressible flows, J. Comput. Phys. 463, 111268 (2022).
B. Xie, and F. Xiao, Toward efficient and accurate interface capturing on arbitrary hybrid unstructured grids: The THINC method with quadratic surface representation and Gaussian quadrature, J. Comput. Phys. 349, 415 (2017).
Z. Sun, S. Inaba, and F. Xiao, Boundary variation diminishing (BVD) reconstruction: A new approach to improve Godunov schemes, J. Comput. Phys. 322, 309 (2016), arXiv: 1602.00814.
F. Xiao, I. Satoshi, and C. Chen, Revisit to the THINC scheme: A simple algebraic VOF algorithm, J. Comput. Phys. 230, 7086 (2011).
X. Deng, B. Xie, R. Loubére, Y. Shimizu, and F. Xiao, Limiter-free discontinuity-capturing scheme for compressible gas dynamics with reactive fronts, Comput. Fluids 171, 1 (2018).
X. Deng, S. Inaba, B. Xie, K. M. Shyue, and F. Xiao, High fidelity discontinuity-resolving reconstruction for compressible multiphase flows with moving interfaces, J. Comput. Phys. 371, 945 (2018).
X. Deng, Y. Shimizu, and F. Xiao, A fifth-order shock capturing scheme with two-stage boundary variation diminishing algorithm, J. Comput. Phys. 386, 323 (2019).
Z. H. Jiang, A higher order interpolation scheme of finite volume method for compressible flow on curvilinear grids, Commun. Comput. Phys. 28, 1609 (2020).
D. Xu, X. Deng, Y. Chen, Y. Dong, and G. Wang, On the freestream preservation of finite volume method in curvilinear coordinates, Comput. Fluids 129, 20 (2016).
P. L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes, J. Comput. Phys. 43, 357 (1981).
M. Vinokur, An analysis of finite-difference and finite-volume formulations of conservation laws, J. Comput. Phys. 81, 1 (1989).
G. S. Jiang, and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202 (1996).
S. Rathan, and N. R. Gande, An improved non-linear weights for seventh-order weighted essentially non-oscillatory scheme, Comput. Fluids 156, 496 (2017).
S. Rathan, N. R. Gande, and A. A. Bhise, Simple smoothness indicator WENO-Z scheme for hyperbolic conservation laws, Appl. Numer. Math. 157, 255 (2020).
G. A. Sod, A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws, J. Comput. Phys. 27, 1 (1978).
P. Wesseling, Principles of Computational Fluid Dynamics (Springer Science and Business Media, Berlin, 2009).
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 51979160, 11902199).
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Jingqi Li and Cheng Liu conceived the present idea and planned the simulations. Jingqi Li developed the theory and carried out the computations. Jingqi Li analyzed the data and verified the numerical methods with the help of Cheng Liu and Changhong Hu. Cheng Liu provided the computing resource. Jingqi Li and Cheng Liu wrote the first draft of the manuscript. Ruoqing Gao and Changhong Hu helped revise and edit the final version. All authors discussed the results and contributed to the final manuscript.
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Li, J., Liu, C., Gao, R. et al. A low-dissipation WENO-THINC scheme for freestream and vortex preservation on general curvilinear grids. Acta Mech. Sin. 39, 322422 (2023). https://doi.org/10.1007/s10409-022-22422-x
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DOI: https://doi.org/10.1007/s10409-022-22422-x