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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
[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
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Chou P Y 1940 Chin. J. Phys. 4 1
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[3] Chou P Y 1945 Quart. Appl. Math. 3 38
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[4] 王圣业, 王光学, 董义道, 邓小刚 2017 物理学报 66 184701
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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
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Wang S Y, Fu X, Yang X L, Zheng H B, Deng X G 2021 Adv. Mech. 51 29
Google Scholar
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[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
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Google Scholar
[10] Menter F R 1994 AIAA J. 32 1598
Google Scholar
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Google Scholar
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Google Scholar
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[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
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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
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Google Scholar
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[29] Strelets M 2001 AIAA Paper 0879
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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
Figure 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Figure 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Figure 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 4 翼型尾迹流网格及切面
Figure 4. Airfoil wake mesh and slices
图 5 翼型尾迹流雷诺切应力分布
Figure 5. Reynolds shear stress distribution on airfoil wake case
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Figure 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 $ Figure 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
Figure 8. NACA0012 airfoil mesh and comparison of surface pressure distributions
图 9 展向涡量云图
Figure 9. Spanwise vorticity distributions
图 10 气动力随时间的变化过程 (a) 升力系数; (b) 阻力系数
Figure 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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