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雷诺应力模型一直是湍流模式理论研究的前沿和难点, 而提高数值鲁棒性是其广泛开展工程应用的关键. 借鉴经典的
$k$ -$kL$ 湍流模型, 本文构造了一种新的${\nu_{\rm{t}}}$ 尺度方程, 并将其用于耦合SSG/LRR模式从而形成SSG/LRR-${\nu_{\rm{t}}}$ 雷诺应力模型. 通过零压力梯度湍流平板边界层、翼型尾迹流、超声速方腔流和NACA0012翼型45°迎角分离流动4个标准算例对新模型进行了验证与确认. 同时, 为了测试模型的数值鲁棒性, 采用高精度数值格式对模型方程进行了离散求解, 并与SA涡粘模型和SSG/LRR-$\omega$ 雷诺应力模型进行对比. 结果表明:${\nu_{\rm{t}}}$ 尺度方程在黏性壁面边界严格为零, 相比传统的$\omega$ 尺度, 具有更好的数值鲁棒性, 从而可实现新模型与高精度数值格式的匹配并获得更好的网格收敛效率; 新模型具备雷诺应力模型的传统优势, 可对拐角流动进行很好的模拟; 具备尺度自适应能力, 对于非定常分离流动的模拟存在一定的潜力.Reynolds stress model has always been the frontier and challenging problem in turbulence model theory research, where improving numerical robustness is the key to its wide application in engineering. Referring to the classical$k$ -$kL$ turbulence model, a new${\nu_t}$ -scale equation is constructed and used to couple the SSG/LRR model to form a so-called SSG/LRR-${\nu_t}$ Reynolds stress model. Four benchmark cases, including zero pressure gradient turbulent plate boundary layer, airfoil wake flow, supersonic square duck flow and separated flow over NACA0012 airfoil at 45 degree angle of attack, are carried out to test the new turbulence model. At the same time, high-order numerical schemes are used to discretize the turbulence equations in order to assess its numerical robustness. The results are compared with those of SA eddy viscosity model and SSG/LRR-$\omega$ Reynolds stress model. It is shown that the${\nu_t} $ -scale equation is strictly equal to zero at the viscous wall boundary. Compared with the traditional$\omega $ -scale, it has better numerical robustness. Along with this, the new model can be matched with the high-order numerical schemes and obtain a better efficiency in the mesh convergence. Moreover, the new model has the inherent advantage of Reynolds stress model in simulating the corner flow and has the potential in scale adaptive simulation of unsteady separated flow.-
Keywords:
- turbulence flow /
- Reynolds stress model /
- separated flow /
- weighted compact nonlinear scheme
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图 1 平板边界层网格收敛性分析 (a) 平板网格示意图; (b) 阻力系数结果
Fig. 1. Convergence analysis on plate boundary layer meshs: (a) Sketch of plate mesh; (b) drag coefficient results
图 2 SSG/LRR-
$ {\nu_{\rm{t}}} $ 模型在$ \Delta y_1^+ = 0.37 $ 网格上的结果 (a)$ x=0.97 $ 处速度型; (b) 摩擦阻力分布Fig. 2. Results of SSG/LRR-
$ {\nu_{\rm{t}}} $ model on grid of$ \Delta y_1^+ = 0.37 $ : (a) u-velocity profile at$ x=0.97 $ ; (b) friction drag coefficient along the plate
图 3
$ x=0.97 $ 处摩擦阻力误差与网格尺度和计算时间的关系 (a) 误差与网格尺度; (b) 误差与计算时间Fig. 3. Relationship between friction drag error at
$ x=0.97 $ and grid scale as well as CPU time: (a) Error vs. grid scale; (b) error vs. CPU time
图 6 方腔流动示意图 (a) 计算网格及边界条件; (b) SA模型在
$ x/D = 50 $ 面上的速度矢量图; (c) SSG/LRR-$ \omega $ 模型在$ x/D = 50 $ 面上的速度矢量图; (d) SSG/LRR-$ {\nu_{\rm{t}}} $ 模型在$ x/D = 50 $ 面上的速度矢量图Fig. 6. Sketch of square duct flow: (a) Mesh and boundary conditions; (b) velocity vector distributions on
$ x/D = 50 $ by SA; (c) velocity vector distributions on$ x/D = 50 $ by SSG/LRR-$ \omega $ ; (d) velocity vector distributions on$ x/D = 50 $ by SSG/LRR-$ {\nu_{\rm{t}}} $ 
图 7 流向速度沿对角线的分布 (a)
$ x/D = 40 $ ; (b)$ x/D = 50 $ Fig. 7. Distribution of u-velocity along diagonal: (a)
$ x/D = 40 $ ; (b)$ x/D = 50 $ 
图 8 NACA0012翼型网格及表面压力分布对比
Fig. 8. NACA0012 airfoil mesh and comparison of surface pressure distributions
图 10 气动力随时间的变化过程 (a) 升力系数; (b) 阻力系数
Fig. 10. History of aerodynamic forces over time: (a) Lift coefficient; (b) Drag coefficient
表 1 再分配项系数
Table 1. Coefficients in redistribution item.
$ {C_1} $ $ {C_1}^* $ $ {C_2} $ $ {C_3} $ $ {C_3}^* $ $ {C_4} $ $ {C_5} $ $ {\phi^\text{(SSG)}} $ 1.7 0.9 1.05 0.8 0.65 0.625 0.2 $ {\phi^\text{(LRR)}} $ 1.8 0 0 0.8 0 0.971 0.578 
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[1] Wilcox D C 2006 Turbulence Modeling for CFD (3rd Ed.) (La Canda: DCW Industries)
[2] 周培源 1940 中国物理学报 4 1
Google Scholar
Chou P Y 1940 Chin. J. Phys. 4 1
Google Scholar
[3] Chou P Y 1945 Quart. Appl. Math. 3 38
Google Scholar
[4] 王圣业, 王光学, 董义道, 邓小刚 2017 物理学报 66 184701
Google Scholar
Wang S Y, Wang G X, Dong Y D, Deng X G 2017 Acta Phys. Sin. 66 184701
Google Scholar
[5] 王圣业, 符翔, 杨小亮, 郑浩榜, 邓小刚 2021 力学进展 51 29
Google Scholar
Wang S Y, Fu X, Yang X L, Zheng H B, Deng X G 2021 Adv. Mech. 51 29
Google Scholar
[6] Slotnick J, Khodadoust A, Alonso J, Darmofal D, Gropp W, Lurie E, Mavriplis D 2014 NASA/CR-2014-218178
[7] Eisfeld B, Brodersen O 2005 AIAA Paper 4727
[8] Launder B E, Reece G L, Rodi W 1975 J. Fluid Mech. 68 537
Google Scholar
[9] Speziale C G, Abid R, Anderson E C 1992 AIAA J. 30 324
Google Scholar
[10] Menter F R 1994 AIAA J. 32 1598
Google Scholar
[11] Eisfeld B, Rumsey C, Togiti V 2016 AIAA J. 54 1524
Google Scholar
[12] Cecora R D, Radespiel R, Eisfeld B, Probst A 2015 AIAA J. 53 739
Google Scholar
[13] Wang S Y, Dong Y D, Deng X G, et al. 2018 J. Aircraft 53 1177
Google Scholar
[14] Eisfeld B, Rumsey C, Togiti V, Braun S, Stürmer A 2021 AIAA J. Published online
[15] Togiti V K, Eisfeld B 2015 AIAA Paper 2925
[16] Ilinca F, Pelletier D 1998 AIAA J. 32 44
[17] Abdol-Hamid K S 2019 AIAA Paper 1878
Google Scholar
[18] 舒博文, 杜一鸣, 高正红, 夏露, 陈树生 2022 航空学报 43 126385
Google Scholar
Shu B W, Du Y M, Gao Z H, Xia L, Chen S S 2022 Acta Aeronautica et Astronautica Sinica 43 126385
Google Scholar
[19] Rotta J C 1968 Aerodynamische Versuchsanstalt Göttingen, Rep. 69 A14
[20] Menter F R, Egorov Y 2010 Flow, Turbulence, and Combustion 85 113
Google Scholar
[21] Abdol-Hamid K S, Carlson J R, Rumsey C L 2016 AIAA Paper 3941
[22] Spalart P R, Allmaras S R 1994 Recherche Aerospatiale 1 5
[23] Speziale C G, Sarkar S, Gatski T B 1991 J 1991 J. Fluid Mech. 227 245
Google Scholar
[24] 王光学, 王圣业, 葛明明, 邓小刚 2018 物理学报 67 194701
Google Scholar
Wang G X, Wang S Y, Ge M M, Deng X G 2018 Acta Phys. Sin. 67 194701
Google Scholar
[25] 李昊, 刘伟, 王圣业 2020 物理学报 69 144702
Google Scholar
Li H, Liu W, Wang S Y 2020 Acta Phys. Sin. 69 144702
Google Scholar
[26] Deng X G, Zhang H X 2000 J. Comput. Phys. 165 90
Google Scholar
[27] Deng X G, Min Y B, Mao M L, Liu H Y, Tu G H, Zhang H X 2013 J. Comput. Phys. 239 90
Google Scholar
[28] Rumsey C Turbulence Modeling Resource https://turbmodels.larc.nasa.gov/
[29] Strelets M 2001 AIAA Paper 0879
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