基于直接数值模拟数据分析的类-1湍流边界层内层律关系
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摘要
湍流边界层普适统计规律探索一直是湍流研究中的重要课题.本文根据Schlatter和?rlü给出的零压梯度平板湍流边界层直接数值模拟数据,对沿流向存在壁面切应力变化的类-1湍流边界层提出再认识的必要性.首次采用时-空间平均摩擦速度作为尺度描述类-1充分发展湍流边界层内层,给出了对内层不依赖于雷诺数基于双控制参数的数学表达,并对其中两个控制参数进行了深刻的物理剖析.指出该数学表达式可以被视为同时考虑和引入了局部壁面切应力增量(Δ)和线性律适用范围特征量(D)这两个关键影响因子修正,并进一步验证了此修正对内层描述的准确性和有效性.研究获得了类-1和类-2湍流边界层内层的统一表达式及其适用范围,发现在大部分情况下,特别是当局部壁面切应力偏离时-空间平均壁面切应力时,内层数学表达式的适用范围在d*=8.0以上,最高可达到d*≈10.0,而传统线性律的适用范围则均在d*=6.0以下.明确揭示了其物理机制是内层速度的表达可根据边界层内层特点,准确描述近壁黏性底层和内层过渡区(非线性增长区)中的速度剖线发展趋势.此机制的发现为湍流边界层内层律的构建和与外层对数律的衔接,并为进一步发展完整统一的类-1、2湍流边界层壁面律提供坚实数学物理基础.
Searching for universal law of turbulent boundary layer(TBL) has been one of the persistent efforts of turbulence research community for the last century. Based on the direct numerical simulation(DNS) data for a zero-pressuregradient flat-plate TBL from Schlatter and ?rlü, the paper pointed out the necessity to reunderstand the conventional TBL theory based on the recent TBL classification proposed by Cao and Xu. Type-A TBL in the classification, as represented by the DNS data, was thoroughly investigated in the inner layer using the velocity scales of both conventional time-averaged local frictional velocity and newly-defined time-space or ensemble-averaged frictional velocity. With the ensemble-averaged frictional velocity as scale, the new mathematical expressions for the inner-layer law were derived by introducing the general damping and enhancing functions. The control parameters in the expressions were found independent on Reynolds number based on the universal governing equation under the ensemble-averaged scales. The physical meanings of the parameters in the law formulation were analyzed and clearly demonstrated that the parameter Δ represented the incremental local wall-shear stress and the parameter D stood for the charateristic length within which the linear viscous law relation was applicable. Comparing to the conventional law-of-the-wall in inner layer, the current law formulation significantly improved the law's predictive accuracy and applicable range. Particularly,when the time-averaged wall-shear stress gets away from the ensmeble-averaged wall shear stress, the new inner-layer law is applicable up to a near-wall range of d*■10.0, while the conventional inner-layer law can only perform well within a wall distance of d*■6.0. With these studies, the properties of the general damping and enhancing functions are well understood, which provides a solid physical and mathematical foundation for developing the complete analytical formulations for law-of-the-wall, which are expected to include the inner layer, the transition layer, the semi-log linear layer and the wake layer.
Searching for universal law of turbulent boundary layer(TBL) has been one of the persistent efforts of turbulence research community for the last century. Based on the direct numerical simulation(DNS) data for a zero-pressuregradient flat-plate TBL from Schlatter and ?rlü, the paper pointed out the necessity to reunderstand the conventional TBL theory based on the recent TBL classification proposed by Cao and Xu. Type-A TBL in the classification, as represented by the DNS data, was thoroughly investigated in the inner layer using the velocity scales of both conventional time-averaged local frictional velocity and newly-defined time-space or ensemble-averaged frictional velocity. With the ensemble-averaged frictional velocity as scale, the new mathematical expressions for the inner-layer law were derived by introducing the general damping and enhancing functions. The control parameters in the expressions were found independent on Reynolds number based on the universal governing equation under the ensemble-averaged scales. The physical meanings of the parameters in the law formulation were analyzed and clearly demonstrated that the parameter Δ represented the incremental local wall-shear stress and the parameter D stood for the charateristic length within which the linear viscous law relation was applicable. Comparing to the conventional law-of-the-wall in inner layer, the current law formulation significantly improved the law's predictive accuracy and applicable range. Particularly,when the time-averaged wall-shear stress gets away from the ensmeble-averaged wall shear stress, the new inner-layer law is applicable up to a near-wall range of d*■10.0, while the conventional inner-layer law can only perform well within a wall distance of d*■6.0. With these studies, the properties of the general damping and enhancing functions are well understood, which provides a solid physical and mathematical foundation for developing the complete analytical formulations for law-of-the-wall, which are expected to include the inner layer, the transition layer, the semi-log linear layer and the wake layer.
引文
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2 von Karman T.Mechanical Similitude and Turbulence.Technical Report.Washington DC:National Advisory Committee on Aeronautics,1931
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5 Khujadze G,Oberlack M.DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow.Theor Comput Fluid Dyn,2004,18:391-411
6 Khujadze G,Oberlack M.DNS and new scaling laws of ZPG turbulent boundary layer flow.In:Proceedings of Fifth International Symposium on Turbulence and Shear Flow Phenomena.July 30-August 2,2019.Southampton,2005.23-26
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9 de Graaff D B,Eaton J K.Reynolds-number scaling of the flat-plate turbulent boundary layer.J Fluid Mech,2000,422:319-346
10 Xu H Y,Guo J H.Direct numerical simulation of turbulence in square and rectangular annular ducts:Part-I mean flow analyses,Reunderstanding fully-developed turbulent boundary layer in duct(in Chinese).Sci Sin-Phys Mech Astron,2013,43:634-651[徐弘一,郭加宏.方型和矩型环管湍流直接数值模拟分析:I.平均流场,对直管道充分发展湍流边界层的再认识.中国科学:物理学力学天文学,2013,43:634-651]
11 Xu H.Direct numerical simulation of turbulence in a square annular duct.J Fluid Mech,2009,621:23-57
12 Cao B,Xu H.Re-understanding the law-of-the-wall for wall-bounded turbulence based on in-depth investigation of DNS data.Acta Mech Sin,2018,34:793-811
13 Schlatter P,?rlüR.Assessment of direct numerical simulation data of turbulent boundary layers.J Fluid Mech,2010,659:116-126
14 Schlatter P,?rlüR,Li Q,et al.Turbulent boundary layers up to Reθ=2500 studied through simulation and experiment.Phys Fluids,2009,21:051702
15 Schlatter P,Li Q,Brethouwer G,et al.Simulations of spatially evolving turbulent boundary layers up to Reθ=4300.Int J Heat Fluid Flow,2010,31:251-261
16 Eitel-Amor G,?rlüR,Schlatter P.Simulation and validation of a spatially evolving turbulent boundary layer up to Reθ=8300.Int J Heat Fluid Flow,2014,47:57-69
17 Krug D,Philip J,Marusic I.Revisiting the law of the wake in wall turbulence.J Fluid Mech,2016,811:421-435
18 Marusic I,McKeon B J,Monkewitz P A,et al.Wall-bounded turbulent flows at high Reynolds numbers:Recent advances and key issues.Phys Fluids,2010,22:065103
19 Wu X,Moin P.Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer.J Fluid Mech,2009,630:5
20 Xu H Y,Cao B C.Research on the near-wall streamwise velocity of fully-developed turbulence in generic straight duct:re-understanding the von Karman Law-of-the-Wall relations(in Chinese).In:National Proceedings of Fluid Dynamics.Beijing,2016.19[徐弘一,曹博超.关于一般直管道充分发展湍流近壁平均流向速度规律讨论:对冯·卡门平壁律的再认知.见:全国流体力学学术会议.北京,2016.19]
21 Liu C,Gao Y,Tian S,et al.Rortex-A new vortex vector definition and vorticity tensor and vector decompositions.Phys Fluids,2018,30:035103,arXiv:1802.04099
2 von Karman T.Mechanical Similitude and Turbulence.Technical Report.Washington DC:National Advisory Committee on Aeronautics,1931
3 Bailey S C C,Hultmark M,Monty J P,et al.Obtaining accurate mean velocity measurements in high Reynolds number turbulent boundary layers using Pitot tubes.J Fluid Mech,2013,715:642-670
4 Mehdi F,Klewicki J C,White C M.Mean momentum balance analysis of rough-wall turbulent boundary layers.Phys D-Nonlin Phenomena,2010,239:1329-1337
5 Khujadze G,Oberlack M.DNS and scaling laws from new symmetry groups of ZPG turbulent boundary layer flow.Theor Comput Fluid Dyn,2004,18:391-411
6 Khujadze G,Oberlack M.DNS and new scaling laws of ZPG turbulent boundary layer flow.In:Proceedings of Fifth International Symposium on Turbulence and Shear Flow Phenomena.July 30-August 2,2019.Southampton,2005.23-26
7 Spalart P R.Direct simulation of a turbulent boundary layer up to Reθ=1410.J Fluid Mech,2006,187:61-98
8 Kline S J,Reynolds W C,Schraub F A,et al.The structure of turbulent boundary layers.J Fluid Mech,1967,30:741-773
9 de Graaff D B,Eaton J K.Reynolds-number scaling of the flat-plate turbulent boundary layer.J Fluid Mech,2000,422:319-346
10 Xu H Y,Guo J H.Direct numerical simulation of turbulence in square and rectangular annular ducts:Part-I mean flow analyses,Reunderstanding fully-developed turbulent boundary layer in duct(in Chinese).Sci Sin-Phys Mech Astron,2013,43:634-651[徐弘一,郭加宏.方型和矩型环管湍流直接数值模拟分析:I.平均流场,对直管道充分发展湍流边界层的再认识.中国科学:物理学力学天文学,2013,43:634-651]
11 Xu H.Direct numerical simulation of turbulence in a square annular duct.J Fluid Mech,2009,621:23-57
12 Cao B,Xu H.Re-understanding the law-of-the-wall for wall-bounded turbulence based on in-depth investigation of DNS data.Acta Mech Sin,2018,34:793-811
13 Schlatter P,?rlüR.Assessment of direct numerical simulation data of turbulent boundary layers.J Fluid Mech,2010,659:116-126
14 Schlatter P,?rlüR,Li Q,et al.Turbulent boundary layers up to Reθ=2500 studied through simulation and experiment.Phys Fluids,2009,21:051702
15 Schlatter P,Li Q,Brethouwer G,et al.Simulations of spatially evolving turbulent boundary layers up to Reθ=4300.Int J Heat Fluid Flow,2010,31:251-261
16 Eitel-Amor G,?rlüR,Schlatter P.Simulation and validation of a spatially evolving turbulent boundary layer up to Reθ=8300.Int J Heat Fluid Flow,2014,47:57-69
17 Krug D,Philip J,Marusic I.Revisiting the law of the wake in wall turbulence.J Fluid Mech,2016,811:421-435
18 Marusic I,McKeon B J,Monkewitz P A,et al.Wall-bounded turbulent flows at high Reynolds numbers:Recent advances and key issues.Phys Fluids,2010,22:065103
19 Wu X,Moin P.Direct numerical simulation of turbulence in a nominally zero-pressure-gradient flat-plate boundary layer.J Fluid Mech,2009,630:5
20 Xu H Y,Cao B C.Research on the near-wall streamwise velocity of fully-developed turbulence in generic straight duct:re-understanding the von Karman Law-of-the-Wall relations(in Chinese).In:National Proceedings of Fluid Dynamics.Beijing,2016.19[徐弘一,曹博超.关于一般直管道充分发展湍流近壁平均流向速度规律讨论:对冯·卡门平壁律的再认知.见:全国流体力学学术会议.北京,2016.19]
21 Liu C,Gao Y,Tian S,et al.Rortex-A new vortex vector definition and vorticity tensor and vector decompositions.Phys Fluids,2018,30:035103,arXiv:1802.04099
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