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High-order discretization of the Reynolds stress model with an εβ-adaptive algorith

雷诺应力模型的εβ自适应高阶离散化

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Abstract

The Reynolds stress model (RSM) outperforms the eddy viscosity model (EVM) when simulating complex flows and has increased demand for high-order discretization. However, the complexity of the RSM equations results in poor numerical stability and weak convergence performance. One of the reasons is that the properties of Reynolds stresses are not fully considered in the design of the numerical scheme. In response to this issue, this study develops an adaptive algorithm to adjust εβ values (an empirical parameter in nonlinear weights) according to the magnitude and smoothness of the Reynolds stresses. This algorithm is introduced into the fifth-order weighted compact nonlinear scheme (WCNS) and is applied to the high-order discretization of the RSM. Three aeronautic test cases are simulated to investigate the performance of the algorithm. The numerical results show that, the adaptive algorithm can reduce the residual by up to 3 orders of magnitude and predict the correct weights for gradient reversals. These results confirm that the application of the εβ-adaptive algorithm to the high-order discretization of the RSM is beneficial both for enhancing convergence and improving resolution.

摘要

雷诺应力模型(RSM)在模拟复杂流动时优于涡流黏性模型(EVM), 且对高阶离散化有更高的要求. 然而由于RSM方程的复杂性, 其数值稳定性差, 收敛性差. 其中一个原因是在设计数值格式时没有充分考虑雷诺应力的特性. 针对这一问题, 本研究开发了一种自适应算法, 根据雷诺应力的大小和平滑度调整εβ值(非线性权重中的经验参数). 该算法被引入到五阶加权紧致非线性格式(WCNS)中, 并应用于RSM的高阶离散化. 通过三个航空测试仿真实例对算法的性能进行了检验. 数值结果表明, 自适应算法可以将残差减少3个数量级, 并预测梯度反转的正确权重. 这表明将εβ自适应算法应用于RSM的高阶离散化有利于提高收敛性和分辨率.

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References

  1. Y. Mor-Yossef, Unconditionally stable time marching scheme for Reynolds stress models, J. Comput. Phys. 276, 635 (2014).

    Article  MathSciNet  Google Scholar 

  2. Y. Liu, Z. Zhou, L. Zhu, and S. Wang, Numerical investigation of flows around an axisymmetric body of revolution by using Reynolds-stress model based hybrid Reynolds-averaged Navier-Stokes/large eddy simulation, Phys. Fluids 33, 085115 (2021).

    Article  Google Scholar 

  3. R. D. Cécora, R. Radespiel, B. Eisfeld, and A. Probst, Differential Reynolds-stress modeling for aeronautics, AIAA J. 53, 739 (2015).

    Article  Google Scholar 

  4. F. Song, and S. B. Pope, Computation of recirculating swirling flow with the GLM Reynolds stress closure, Acta Mech. Sin. 10, 110 (1994).

    Article  Google Scholar 

  5. B. Eisfeld, V. Togiti, S. Braun, and A. W. Stuermer, in Reynolds-stress model computations of the NASA juncture flow experiment: Proceedings of AIAA Scitech 2020 Forum (Orlando, 2020).

  6. J. Slotnick, A. Khodadoust, J. Alonso, D. Darmofal, W. Gropp, E. Lurie, and D. J Mavriplis, CFD vision 2030 study: a path to revolutionary computational aerosciences (NASA Langley Research Center, Hampton, 2014).

    Google Scholar 

  7. Z. J. Wang, K. Fidkowski, R. Abgrall, F. Bassi, D. Caraeni, A. Cary, H. Deconinck, R. Hartmann, K. Hillewaert, H. T. Huynh, N. Kroll, G. May, P. O. Persson, B. van Leer, and M. Visbal, High-order CFD methods: current status and perspective, Int. J. Numer. Meth. Fluids 72, 811 (2013).

    Article  MathSciNet  Google Scholar 

  8. C. Zhou, and Z. J. Wang, in CPR high-order discretization of the RANS equations with the SA model: Proceedings of 53rd AIAA Aerospace Sciences Meeting (Kissimmee, 2015).

  9. M. Tiberga, A. Hennink, J. L. Kloosterman, and D. Lathouwers, A high-order discontinuous Galerkin solver for the incompressible RANS equations coupled to the k-ε turbulence model, Comput. Fluids 212, 104710 (2020).

    Article  MathSciNet  Google Scholar 

  10. X. Yang, J. Cheng, H. Luo, and Q. Zhao, Robust implicit direct discontinuous galerkin method for simulating the compressible turbulent flows, AIAA J. 57, 1113 (2019).

    Article  Google Scholar 

  11. C. Sheng, Improving predictions of transitional and separated flows using RANS modeling, Aerosp. Sci. Tech. 106, 106067 (2020).

    Article  Google Scholar 

  12. Y. G. Lai, Computational method of second-moment turbulence closures in complex geometries, AIAA J. 33, 1426 (1995).

    Article  Google Scholar 

  13. U. Schumann, Realizability of Reynolds-stress turbulence models, Phys. Fluids 20, 721 (1977).

    Article  Google Scholar 

  14. N. Ben Nasr, G. A. Gerolymos, and I. Vallet, Low-diffusion approximate Riemann solvers for Reynolds-stress transport, J. Comput. Phys. 268, 186 (2014), arXiv: 1307.2154.

    Article  MathSciNet  Google Scholar 

  15. S. Wang, Y. Dong, X. Deng, G. Wang, and J. Wang, High-order simulation of aeronautical separated flows with a Reynold stress model, J. Aircraft 55, 1177 (2018).

    Article  Google Scholar 

  16. X. D. Liu, S. Osher, and T. Chan, Weighted essentially non-oscillatory schemes, J. Comput. Phys. 115, 200 (1994).

    Article  MathSciNet  Google Scholar 

  17. X. Deng, and M. Mao, in Weighted compact high-order nonlinear schemes for the Euler equations: Proceedings of 13th Computational Fluid Dynamics Conference (Snowmass Village, 1997).

  18. Y. Shen, G. Zha, and B. Wang, Improvement of stability and accuracy for weighted essentially nonoscillatory scheme, AIAA J. 47, 331 (2009).

    Article  Google Scholar 

  19. S. Zheng, X. Deng, D. Wang, and C. Xie, A parameter-free ε-adaptive algorithm for improving weighted compact nonlinear schemes, Int. J. Numer. Meth. Fluids 90, 247 (2019).

    Article  MathSciNet  Google Scholar 

  20. N. K. Yamaleev, and M. H. Carpenter, Third-order energy stable WENO scheme, J. Comput. Phys. 228, 3025 (2009).

    Article  MathSciNet  Google Scholar 

  21. W. S. Don, and R. Borges, Accuracy of the weighted essentially nonoscillatory conservative finite difference schemes, J. Comput. Phys. 250, 347 (2013).

    Article  MathSciNet  Google Scholar 

  22. F. Arándiga, M. C. Martí, and P. Mulet, Weights design for maximal order WENO schemes, J. Sci. Comput. 60, 641 (2014).

    Article  MathSciNet  Google Scholar 

  23. F. Jia, Z. Gao, and W. S. Don, A spectral study on the dissipation and dispersion of the WENO schemes, J. Sci. Comput. 63, 49 (2015).

    Article  MathSciNet  Google Scholar 

  24. Z. He, Y. Zhang, X. Li, L. Li, and B. Tian, Preventing numerical oscillations in the flux-split based finite difference method for compressible flows with discontinuities, J. Comput. Phys. 300, 269 (2015).

    Article  MathSciNet  Google Scholar 

  25. D. C. Wilcox, Turbulence Modeling for CFD, 3rd ed. (DCW Industries, La Canada, 2006).

    Google Scholar 

  26. F. R. Menter, Two-equation eddy-viscosity turbulence models for engineering applications, AIAA J. 32, 1598 (1994).

    Article  Google Scholar 

  27. X. Deng, and H. Zhang, Developing high-order weighted compact nonlinear schemes, J. Comput. Phys. 165, 22 (2000).

    Article  MathSciNet  Google Scholar 

  28. B. van Leer, Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method, J. Comput. Phys. 32, 101 (1979).

    Article  Google Scholar 

  29. X. Deng, Y. Min, M. Mao, H. Liu, G. Tu, and H. Zhang, Further studies on Geometric Conservation Law and applications to high-order finite difference schemes with stationary grids, J. Comput. Phys. 239, 90 (2013).

    Article  MathSciNet  Google Scholar 

  30. X. Deng, X. Liu, M. Mao, H. Zhang, in Investigation on weighted compact fifth-order nonlinear scheme and applications to complex flow: 17th AIAA Computational Fluid Dynamics Conference (Toronto, 2005).

  31. G. S. Jiang, and C. W. Shu, Efficient implementation of weighted ENO schemes, J. Comput. Phys. 126, 202 (1996).

    Article  MathSciNet  Google Scholar 

  32. Turbulence Modeling Resource, (2020). https://turbmodels.larc.nasa.gov.

  33. A. Nakayama, Characteristics of the flow around conventional and supercritical airfoils, J. Fluid Mech. 160, 155 (1985).

    Article  Google Scholar 

  34. J. Barche, T. W. Binion, K. G. Winter, L. H. Ohman, J. Sloof, and P. J. Bobbitt, Experimental data base for computer program assessment, NATO Advisory Group for Aerospace Research & Development, Neuilly-sur-Seine (1979).

  35. AIAA CFD Drag Prediction Workshop, (2015). https://aiaa-dpw.larc.nasa.gov/Workshop5/workshop5.html.

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Authors and Affiliations

  1. College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, 410073, China

    Xiang Fu  (符翔), Xiaogang Deng  (邓小刚), Shengye Wang  (王圣业), Shichao Zheng  (郑世超) & Guangxue Wang  (王光学)

  2. Chinese Academy of Military Science, Beijing, 100071, China

    Xiaogang Deng  (邓小刚)

  3. School of Physics, Sun Yat-sen University, Guangzhou, 510275, China

    Guangxue Wang  (王光学)

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  1. Xiang Fu  (符翔)
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  2. Xiaogang Deng  (邓小刚)
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  3. Shengye Wang  (王圣业)
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  4. Shichao Zheng  (郑世超)
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  5. Guangxue Wang  (王光学)
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Corresponding author

Correspondence to Shengye Wang  (王圣业).

Additional information

This work was supported by the National Natural Science Foundation of China (Grant Nos. 12002379 and 11972370) and the National Key Project (Grant No. GJXM92579).

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Fu, X., Deng, X., Wang, S. et al. High-order discretization of the Reynolds stress model with an εβ-adaptive algorith. Acta Mech. Sin. 38, 321357 (2022). https://doi.org/10.1007/s10409-021-09084-x

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