湍流中的多态现象
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摘要
湍流是自然界最普遍的流动现象之一,被称为是经典力学中最后的难题之一.经典的湍流理论都假设湍流是各态遍历的,也就是说虽然瞬时流场的性质可能受初始流场影响很大,但是其统计平均量却是不依赖于初始流场的.各态遍历理论是湍流理论与建模的基础.然而,近些年一些实验和数值模拟的结果表明:部分湍流问题里存在多湍流态现象,即在同一组参数下可能存在不唯一的统计结果和流动结构.本文回顾已经报道过多态现象的Rayleigh-Bénard对流(Rayleigh-Bénardconvection,RBC)、 vonKármán流动(vonKármánflow,VKF)、Taylor-Couette流动(Taylor-Couette flow, TCF)、球形Couette流动(spherical Couette flow, SCF)、Taylor-Green驱动下的旋转湍流(rotating homogeneous turbulence with Taylor-Green forcing, TGF)以及作者研究的展向旋转平板Couette流动(rotating plane Couette flows, RPCF).在每个问题上,都将一一介绍流动问题的定义、多态现象出现的条件、形态等.在这些多湍流态问题里,有一些流动是由于不同的初始流场引起的不同流态(例如VKF,TCF,SCF,TGF和RPCF);而另一些流动则是在同一种初始条件下,随着时间的推移,流动会在不同流态之间自发切换(例如RBC).最后,还对湍流中的多态问题展开一些讨论和展望.
Turbulence is ubiquitous in nature, and is known as one of the unresolved problems in classical physics. The classical theory of turbulence assumed that turbulence is ergodic, which means that when turbulence is in a stationary state, even though the instantaneous properties are sensitive to initial conditions, the statistical averages of the instantaneous properties, such as the mean profiles and the skin friction in wall-bounded turbulence, are unique at any fixed set of parameters. This classical ergodic theory is the foundation of turbulence theory and modeling, and makes it possible to extrapolate data from lower Reynolds number turbulence to higher ones. However, in recent years, a few experimental and numerical evidences showed that multiple states exist in several flow problems. That is, the turbulent statistics and flow structures are not the same even at the same control parameters. In this paper, several published flow problems with multiple states are reviewed, including Rayleigh-Bénard convection(RBC), von Kármán flow(VKF), Taylor-Couette flow(TCF), spherical Couette flow(SCF), rotating homogeneous turbulence with Taylor-Green forcing(TGF), and the spanwise rotating plane Couette flows(RPCF). Two different categories of multiple states can be observed from these flow problems. One is in RBC, where several research groups found that the system will have different flow states and it will switch between different states as time evolves. The other can be generally seen from other five flows, where initial conditions or hysteresis effect can be observed. For example, hysteresis loops were reported in the experiments of TCF for global torque and local velocities at very high Reynolds numbers, while in RPCF, different flow statistics and flow structures were obtained with different initial flow fields based on the same code, the same computational domain and the same grid resolution at the same control parameters. Whether the second type of multiple states will finally turn into the first one in a long enough time is an open question which deserves further investigations. The underlying mechanism of multiple states is not clear at the present moment, which also demands continued effort and studies. In the six flow problems mentioned above, large-scale flow structures persist, which may highlight the importance of the coherent structures and their selectability in turbulence, as concluded by Huisman et al. Another possible guess from myself is the multiple competing flow mechanisms in the flow problems with multiple states. For example, in rotating homogeneous turbulence with Taylor-Green forcing(TGF), there are two competing mechanisms, one is the viscosity dissipation which causes the forward cascade, the other is the system rotation which results in the inverse cascade. The two competing mechanisms will make the system has more equilibrium points and multiple states. I believe that there will be more and more flow problems with multiple states reported in the future, together with a better understanding of the underling mechanism.
Turbulence is ubiquitous in nature, and is known as one of the unresolved problems in classical physics. The classical theory of turbulence assumed that turbulence is ergodic, which means that when turbulence is in a stationary state, even though the instantaneous properties are sensitive to initial conditions, the statistical averages of the instantaneous properties, such as the mean profiles and the skin friction in wall-bounded turbulence, are unique at any fixed set of parameters. This classical ergodic theory is the foundation of turbulence theory and modeling, and makes it possible to extrapolate data from lower Reynolds number turbulence to higher ones. However, in recent years, a few experimental and numerical evidences showed that multiple states exist in several flow problems. That is, the turbulent statistics and flow structures are not the same even at the same control parameters. In this paper, several published flow problems with multiple states are reviewed, including Rayleigh-Bénard convection(RBC), von Kármán flow(VKF), Taylor-Couette flow(TCF), spherical Couette flow(SCF), rotating homogeneous turbulence with Taylor-Green forcing(TGF), and the spanwise rotating plane Couette flows(RPCF). Two different categories of multiple states can be observed from these flow problems. One is in RBC, where several research groups found that the system will have different flow states and it will switch between different states as time evolves. The other can be generally seen from other five flows, where initial conditions or hysteresis effect can be observed. For example, hysteresis loops were reported in the experiments of TCF for global torque and local velocities at very high Reynolds numbers, while in RPCF, different flow statistics and flow structures were obtained with different initial flow fields based on the same code, the same computational domain and the same grid resolution at the same control parameters. Whether the second type of multiple states will finally turn into the first one in a long enough time is an open question which deserves further investigations. The underlying mechanism of multiple states is not clear at the present moment, which also demands continued effort and studies. In the six flow problems mentioned above, large-scale flow structures persist, which may highlight the importance of the coherent structures and their selectability in turbulence, as concluded by Huisman et al. Another possible guess from myself is the multiple competing flow mechanisms in the flow problems with multiple states. For example, in rotating homogeneous turbulence with Taylor-Green forcing(TGF), there are two competing mechanisms, one is the viscosity dissipation which causes the forward cascade, the other is the system rotation which results in the inverse cascade. The two competing mechanisms will make the system has more equilibrium points and multiple states. I believe that there will be more and more flow problems with multiple states reported in the future, together with a better understanding of the underling mechanism.
引文
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3 Kolmogorov A N.The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Dokl Akad Nauk SSSR,1941,30:301-305
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5 Kachanov Y S.Physical mechanisms of laminar-boundary-layer transition.Annu Rev Fluid Mech,1994,26:411-482
6 Lee C B,Wu J Z.Transition in wall-bounded flows.Appl Mech Rev,2008,61:030802
7 Andereck C D,Liu S S,Swinney H L.Flow regimes in a circular Couette system with independently rotating cylinders.J Fluid Mech,1986,164:155-183
8 Martinez-Arias B,Peixinho J,Crumeyrolle O,et al.Effect of the number of vortices on the torque scaling in Taylor-Couette flow.J Fluid Mech,2014,748:756-767
9 Eckhardt B,Schneider T M,Hof B,et al.Turbulence transition in pipe flow.Annu Rev Fluid Mech,2007,39:447-468
10 Herbert T.Secondary instability of boundary layers.Annu Rev Fluid Mech,1988,20:487-526
11 Xi H D,Xia K Q.Flow mode transitions in turbulent thermal convection.Phys Fluids,2008,20:055104
12 van der Poel E P,Stevens R A J M,Lohse D.Connecting flow structures and heat flux in turbulent Rayleigh-Benard convection.Phys Rev E,2011,84:045303
13 Ahlers G,Funfschilling D,Bodenschatz E.Heat transport in turbulent Rayleigh-Benard convection for Pr?0.8 and Ra≤1015.J Phys Conf Ser,2011,318:082001
14 Grossmann S,Lohse D.Multiple scaling in the ultimate regime of thermal convection.Phys Fluids,2011,23:045108
15 Weiss S,Ahlers G.Effect of tilting on turbulent convection:Cylindrical samples with aspect ratioλ=0.50.J Fluid Mech,2013,715:314-334
16 Xie Y C,Ding G Y,Xia K Q.Flow topology transition via global bifurcation in thermally driven turbulence.Phys Rev Lett,2018,120:214501
17 Ravelet F,Marie L,Chiffaudel A,et al.Multistability and memory effect in a highly turbulent flow:Experimental evidence for a global bifurcation.Phys Rev Lett,2004,93:164501
18 Ravelet F,Chiffaudel A,Daviaud F.Supercritical transition to turbulence in an inertially driven von Karman closed flow.J Fluid Mech,2008,601:339-364
19 Cortet P P,Chiffaudel A,Daviaud F,et al.Experimental evidence of a phase transition in a closed turbulent flow.Phys Rev Lett,2010,105:214501
20 Huisman S G,van der Veen R C A,Sun C,et al.Multiple states in highly turbulent Taylor-Couette flow.Nat Commun,2014,5:3820
21 van del Veen R C A,Huisman S G,Dung O Y,et al.Exploring the phase space of multiple states in highly turbulent Taylor-Couette flows.Phys Rev Fluids,2016,1:024401
22 Grossmann S,Lohse D,Sun C.High-Reynolds number Taylor-Couette turbulence.Annu Rev Fluid Mech,2016,48:53-80
23 Gul M,Elsinga G E,Westerweel J.Experimental investigation of torque hysteresis behavior of Taylor-Couette flow.J Fluid Mech,2018,836:635-648
24 Zimmerman D S,Triana S A,Lathrop D P.Bi-stability in turbulent,rotating spherical Couette flow.Phys Fluids,2011,23:065104
25 Yokoyama N,Takaoka M.Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence.Phys Rev Fluids,2017,2:092602
26 Xia Z H,Shi Y P,Cai Q D,et al.Multiple states in turbulent plane Couette flow with spanwise rotation.J Fluid Mech,2018,837:477-490
27 Huang Y H,Xia Z H,Wan M P,et al.Numerical investigation of plane Couette flow with weak spanwise rotation.Sci China Phys Mech Astro,2019,62:044711
28 Xi H D,Lam S,Xia K Q.From laminar plumes to organized flows:The onset of large-scale circulation in turbulent thermal convection.JFluid Mech,2004,503:47-56
29 Stevens R J A M,Clercx H J H,Lohse D.Breakdown of the large-scale circulation inΓ=1/2 rotating Rayleigh-Benard flow.Phys Rev E,2012,86:056311
30 Gai J,Xia Z H,Cai Q D,et al.Turbulent statistics and flow structures in spanwise-rotating turbulent plane Couette flows.Phys Rev Fluids,2016,1:054401
31 Bech K H,Andersson H I.Secondary flow in weakly rotating turbulent plane Couette flow.J Fluid Mech,1996,317:195-214
32 Wang P,Hsieh D Y,Tang S Q,et al.Pattern selection in a reaction-diffusion equation.Acta Mech Sin,2002,18:652-660
33 Hristov G,Ansell P J.Poststall hysteresis and flow field unsteadiness on a NACA0012 airfoil.AIAA J,2018,56:2528-2539
2 Reynolds O.An experimental investigation of the circumstances which determine whether the motion of water shall be direct or sinuous,and of the law of resistance in parallel channels.Proc Royal Soc London,1883,35:84-99
3 Kolmogorov A N.The local structure of turbulence in incompressible viscous fluid for very large Reynolds numbers.Dokl Akad Nauk SSSR,1941,30:301-305
4 Kolmogorov A N.On degenaration of isotropic turbulence in an incompressible viscous liquid.Dokl Akad Nauk URSS,1941,30:538-540
5 Kachanov Y S.Physical mechanisms of laminar-boundary-layer transition.Annu Rev Fluid Mech,1994,26:411-482
6 Lee C B,Wu J Z.Transition in wall-bounded flows.Appl Mech Rev,2008,61:030802
7 Andereck C D,Liu S S,Swinney H L.Flow regimes in a circular Couette system with independently rotating cylinders.J Fluid Mech,1986,164:155-183
8 Martinez-Arias B,Peixinho J,Crumeyrolle O,et al.Effect of the number of vortices on the torque scaling in Taylor-Couette flow.J Fluid Mech,2014,748:756-767
9 Eckhardt B,Schneider T M,Hof B,et al.Turbulence transition in pipe flow.Annu Rev Fluid Mech,2007,39:447-468
10 Herbert T.Secondary instability of boundary layers.Annu Rev Fluid Mech,1988,20:487-526
11 Xi H D,Xia K Q.Flow mode transitions in turbulent thermal convection.Phys Fluids,2008,20:055104
12 van der Poel E P,Stevens R A J M,Lohse D.Connecting flow structures and heat flux in turbulent Rayleigh-Benard convection.Phys Rev E,2011,84:045303
13 Ahlers G,Funfschilling D,Bodenschatz E.Heat transport in turbulent Rayleigh-Benard convection for Pr?0.8 and Ra≤1015.J Phys Conf Ser,2011,318:082001
14 Grossmann S,Lohse D.Multiple scaling in the ultimate regime of thermal convection.Phys Fluids,2011,23:045108
15 Weiss S,Ahlers G.Effect of tilting on turbulent convection:Cylindrical samples with aspect ratioλ=0.50.J Fluid Mech,2013,715:314-334
16 Xie Y C,Ding G Y,Xia K Q.Flow topology transition via global bifurcation in thermally driven turbulence.Phys Rev Lett,2018,120:214501
17 Ravelet F,Marie L,Chiffaudel A,et al.Multistability and memory effect in a highly turbulent flow:Experimental evidence for a global bifurcation.Phys Rev Lett,2004,93:164501
18 Ravelet F,Chiffaudel A,Daviaud F.Supercritical transition to turbulence in an inertially driven von Karman closed flow.J Fluid Mech,2008,601:339-364
19 Cortet P P,Chiffaudel A,Daviaud F,et al.Experimental evidence of a phase transition in a closed turbulent flow.Phys Rev Lett,2010,105:214501
20 Huisman S G,van der Veen R C A,Sun C,et al.Multiple states in highly turbulent Taylor-Couette flow.Nat Commun,2014,5:3820
21 van del Veen R C A,Huisman S G,Dung O Y,et al.Exploring the phase space of multiple states in highly turbulent Taylor-Couette flows.Phys Rev Fluids,2016,1:024401
22 Grossmann S,Lohse D,Sun C.High-Reynolds number Taylor-Couette turbulence.Annu Rev Fluid Mech,2016,48:53-80
23 Gul M,Elsinga G E,Westerweel J.Experimental investigation of torque hysteresis behavior of Taylor-Couette flow.J Fluid Mech,2018,836:635-648
24 Zimmerman D S,Triana S A,Lathrop D P.Bi-stability in turbulent,rotating spherical Couette flow.Phys Fluids,2011,23:065104
25 Yokoyama N,Takaoka M.Hysteretic transitions between quasi-two-dimensional flow and three-dimensional flow in forced rotating turbulence.Phys Rev Fluids,2017,2:092602
26 Xia Z H,Shi Y P,Cai Q D,et al.Multiple states in turbulent plane Couette flow with spanwise rotation.J Fluid Mech,2018,837:477-490
27 Huang Y H,Xia Z H,Wan M P,et al.Numerical investigation of plane Couette flow with weak spanwise rotation.Sci China Phys Mech Astro,2019,62:044711
28 Xi H D,Lam S,Xia K Q.From laminar plumes to organized flows:The onset of large-scale circulation in turbulent thermal convection.JFluid Mech,2004,503:47-56
29 Stevens R J A M,Clercx H J H,Lohse D.Breakdown of the large-scale circulation inΓ=1/2 rotating Rayleigh-Benard flow.Phys Rev E,2012,86:056311
30 Gai J,Xia Z H,Cai Q D,et al.Turbulent statistics and flow structures in spanwise-rotating turbulent plane Couette flows.Phys Rev Fluids,2016,1:054401
31 Bech K H,Andersson H I.Secondary flow in weakly rotating turbulent plane Couette flow.J Fluid Mech,1996,317:195-214
32 Wang P,Hsieh D Y,Tang S Q,et al.Pattern selection in a reaction-diffusion equation.Acta Mech Sin,2002,18:652-660
33 Hristov G,Ansell P J.Poststall hysteresis and flow field unsteadiness on a NACA0012 airfoil.AIAA J,2018,56:2528-2539