壁湍流的展向运动减阻机理研究
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摘要
运动物体的表面通常存在湍流边界层,以条带和流向涡为主要特征的壁湍流拟序结构是壁面摩擦阻力的主要来源。通过主动控制技术对湍流近壁区域流动的展向方向进行修正,可以有效干扰湍流拟序结构,抑制壁湍流的产生,达到良好的减阻效果。本文以槽道流形成的充分发展的湍流边界层为基本研究对象,利用傅里叶—契比雪夫谱方法,对壁湍流的多种展向运动控制方法进行了直接数值模拟(DNS),并建立了相应的数据库对湍流减阻规律进行了分析。控制方法可分为两类：一类是展向运动壁面,包括展向振动壁面和展向振动、流向传播的波动壁面；另一类是展向运动电磁力,包括展向行波电磁力和展向振荡、流向传播的波动电磁力。数值模拟时,流场的基本方程为不可压缩牛顿流体的Navier-Stokes方程(N-S方程)。根据控制方法的不同,对N-S方程添加不同的边界条件或者源项。
1)对展向振动壁面的流动控制和减阻问题进行了研究。通过改变振幅大小和振动周期,可以使壁面摩擦阻力明显减少。随着平均减阻率的增加,壁面阻力随时间的变化也更加稳定并呈现出较大周期的变化。将一个典型的阻力变化周期分成三个特征时段,对湍流脉动能的波谱、频谱变化规律以及展向振动壁面抑制壁湍流、实现减阻的内在机理进行了分析。结果表明,总涡能在得到不同程度抑制的同时,三个特征时段相关物理量的谱具有不同的变化特点。从二维能谱能量的分布上看,谱能在整体下降的同时,在波数相对较低的区域,能量有向某一波矢方向集中的趋势。结合近壁湍流拟序结构的变化规律,揭示出展向振动壁面减阻是两种机理共同作用的结果,特定时段由某一机理起主导作用。
2)对展向振动、流向传播的波动壁面的流动控制和减阻问题进行了研究。讨论了流向波动参数(波数)对相关统计量以及壁面阻力的影响。根据诱导边界层(广义Stokes层)调制下近壁湍流拟序结构以及湍流猝发事件的变化规律,对此类波动壁面的湍流控制和减阻机理进行了分析。结果表明,当此类波动壁面被用来调制近壁流动时,固有的湍流流场和诱导流场会各自发生变化。仅低频波对湍流流场具有显著影响,可导致近壁湍流拟序结构以及相应的湍流猝发事件的显著变化；波数的增大对于湍流猝发事件的频率和强度增减的影响并不同步,存在一个最优的波数,在其调制下,固有流场对诱导流场的影响最弱,而诱导流场对固有流场的影响显著,减阻效果最好。此外,给出了最优减阻率随振幅及振动周期的变化曲线,对具有最佳减阻效果的最优控制波数的存在进行了确认。
3)对展向行波电磁力(STW)的湍流控制和减阻机理进行了研究。结果表明,展向行波电磁力对湍流的控制过程实质上是一种由电磁力诱导的特殊流场对壁湍流的调制过程。调制流场由一系列的沿流向伸展并沿展向传播的,距离相等、相互平行的负流向涡组成。该流向涡可以减弱湍流固有的正流向涡和兼并湍流固有的负流向涡。于是,在这种反复周期出现的调制波作用下,湍流流场的流向涡结构和条带数逐渐减少,调制流场也因此逐渐主宰壁面边界层,使其层流化发展,这导致了壁面阻力的下降。优化参数控制下,平均减阻率最高可达30%以上。此外,讨论了展向行波电磁力的波长对控制效果的影响,结果表明,对于长波电磁力,固有流场对诱导流场的影响较弱,而诱导流场可以有效地减少固有流场的“条带—涡”结构,抑制湍流猝发活动,故减阻效果理想；对于短波电磁力,诱导流场作用下,近壁湍流上扬、下扫活动明显增强,故增阻效果显著。
4)研究了展向振荡、流向传播的波动电磁力的流动控制和减阻问题。讨论了流向波数对壁面阻力及相关统计量的影响。从展向速度、近壁拟序结构以及湍流猝发事件的频率和强度等方面对其控制壁湍流、实现壁面减阻的内在机理进行了分析。结果表明,当此类波动电磁力被用来调制近壁流动时,控制效果和展向振动、流向传播的波动壁面有许多相似之处,仅低频波对湍流流场具有显著影响,可导致湍流猝发事件的频率和强度的显著变化；波数的增大对于湍流猝发事件的频率和强度增减的影响并不一致,存在一个最佳的控制波数,在其调制下,减阻效果最好。此外,本文的计算结果显示,波动电磁力的振动参数对最佳控制波数和最优减阻率的影响较为显著。优化参数控制下,展向振荡、流向传播的波动电磁力的能量利用效率要高于振荡电磁力。(此部分内容被国际著名物理类期刊Physics of Fluids审稿专家评为The work is well motivated and the results are interesting. They provide new and relevant information on the effect of an oscillating spanwise Lorentz force on a turbulent channel flow. In particular, the effect of a streamwise wavenumber in the force distribution is studied. Overall, I think the manuscript will merit publication in Phyiscs of Fluids.)
The surface of most mobile devices is inwrapped by turbulent bundary layer. The coherent structures of wall turbulence, charactered by the streaks and longitudinal vortex structures, are responsible for the generation of high turbulent skin-friction drag. When spanwise motions imposed by external forcing excitation are applied to modify the near-wall turbulent flow, the coherent structure of wall turbulence can be disrupted effectively, thereby suppressing the production of turbulence near the wall, leading to significant turbulent skin-friction drag reduction. Several different ways of introducing spanwise flow modifications in turbulent channel flow are investigated by direct numerical simulation (DNS) based on standard Fourier-Chebyshev spectral method. The database of turbulent channel flow is produced by DNS. Control strategies can be classified into two categories. The first class uses wall motions, including spanwise wall oscillation, and streamwise travelling wave induced by spanwise wall oscillation; the second class adopts Lorentz forcing exitation, including spanwise and streamwise travelling waves induced by spanwise oscillating Lorentz force. In the simulations, the incompressible Navier-Stokes (N-S) equations are used as the basic control equations to describe the turbulent channel flow. Different boundary conditions or body-force terms are added to the N-S equations according to the adopted control strategies.
1) Flow control and drag reduction due to spanwise wall oscillation in turbulent channel flow are investigated numerically. The skin-friction drag can be reduced significantly by changing the amplitude and the period of the spanwise wall oscillation. Along with the increase of the mean drag reduction rate, time evolution of skin-friction drag becomes steadier and shows longer periodicity. A type drag reduction periodicity is divided to three typical phases to investigate the energy spectra of turbulent fluctuation kinetic energy and flow physics of turbulent channel flow subject to spanwise wall oscillation. It is found that the suppressions of total vorticity energy for these three phases are different. The distribution of the two-dimensional energy spectra of velocity fluctuation show that turbulent fluctuation kinetic energy decrease significantly, and there seems a trend that energy concentrates to a certain wave vector at low-wave number end. Combining these observations with the different states of the near-wall turbulent structures in the three typical phases, the mechanisms of turbulence suppression and drag reduction via spanwise wall oscillation are further analysed. The results suggest that two kinds of mechanisms of turbulence suppression and drag reduction due to spanwise wall oscillation work by turns, one of which is the incline of streamwise vorticity and the streaks induced by oscillation of the wall, leading to creation of a negative spanwise vorticity in the near-wall region, which is in favor of drag reduction, and the other is the relative displacement of the streamwise vorticity and the streaks over the oscillating wall, which can induce the streaks broad and the streamwise vorticity weak.
2) The control and drag reduction in turbulent channel flow via streamwise travelling wave induced by spanwise-wall oscillation are investigated by DNS. The effects of streamwise wave number on the the associated statistics and the skin-friction drag are discussed. The induced Stokes layer, the near-wall flow structure, as well as the turbulent burst events are analysed. In addition, the mechanisms of turbulence suppression and drag reduction via streamwise traveling wave induced by spanwise-wall oscillation are also discussed. The results suggest that the intrinsic and induced flows are modified each other when the travelling wavy wall is utilized to modulate the near-wall turbulent flow. Only the low-frequency waves have the significant influence on the near-wall turbulent structures which induces the great variations of the the turbulent bursting events which can be detected by VITA detective technique. The variations of the frequency and the intensity of the burst are unsynchronized with wave number, and there exists an optimal wave number, in which the influence of the intrinsic flow on the induced flow is weakest, and on the contrary, the intrinsic turbulent flow is modified strongly by the induced flow, and the largest amount of drag reduction is obtained. Otherwise, the effects of the oscillation parameters on the maximum drag reduction rate are presented, further confirming the existence of the optimal wave number for drag reduction.
3) Mechanism for turbulent control and drag reduction in a channel flow subject to a spanwise travelling wave (STW) via Lorentz forcing is investigated numerically. The results show that the intrinsic flow structures are modified by the induced flow on application of STW control to the turbulent bundary layer. The induced flow is consisted by a distinct set of longitudinal vortices, moving along the spanwisae direction with same spanwise spacing, and paralleling with each other. As travelling with the spanwise Lorentz force, these generated longitudinal vortices can suppress positive random longitudinal vortices and merge negative random longitudinal vortices in the near-wall region of the intrinsic turbulent flow. Then, the intrinsic longitudinal vortex structures and the near-wall pairs of streaks are reduced over time. The induced flow finally dominates the near-wall flow and consequently relaminarizes the flow, leading to a reduction in turbulent drag of more than30%. Otherwise, the influences of the spanwise wavelength in the control were discussed. As the wavelength large enough, the influence of the intrinsic flow on the induced flow is weaker, and on the contrary, the streaks and vortex structures in the intrinsic turbulent flow can be reduced remarkably by the induced flow, and correspondingly, the turbulent bursting events are suppressed efficiently. Consequently, larger amount of drag reduction is obtained. If the wavelength is too short, ejection and sweeping activities of the fluid will be enhanced significantly by the induced flow, which will result in notable drag increase.
4) The control and drag reduction in turbulent channel flow via streamwise travelling wave induced by spanwise oscillatory Lorentz force are investigated numerically by DNS. The effects of streamwise control parameter on the skin-friction drag and associated statistics are discussed. The induced spanwise velocity, near-wall flow structures, as well as the frequency and intensity of the turbulent bursting events are analysed to disclose the mechanisms of turbulence suppression and drag reduction via such wavy Lorentz force. The results are very similar with what have been obtained by the streamwise travelling wave induced by spanwise-wall oscillation. Only the low-frequency waves have the significant influence on the near-wall turbulent structures which induces the great variations of the the turbulent bursting events. The results show that the frequency and intensity vary with wave number in the contradictious tendencies. There is an optimal wave number in flow control, in which the maximum is obtained. The optimal wave number and the maximum drag reduction are also affected by the oscillation parameters of the wavy Lorentz forces. In addition, the results show that there is an efficiency improvement for the proposed method compared with the spanwise oscillating Lorentz force method.(The reviewers of Physics of Fluids appraise that the work is well motivated and the results are interesting. They provide new and relevant information on the effect of an oscillating spanwise Lorentz force on a turbulent channel flow. In particular, the effect of a streamwise wavenumber in the force distribution is studied. Overall, I think the manuscript will merit publication in Phyiscs of Fluids.)
1)对展向振动壁面的流动控制和减阻问题进行了研究。通过改变振幅大小和振动周期,可以使壁面摩擦阻力明显减少。随着平均减阻率的增加,壁面阻力随时间的变化也更加稳定并呈现出较大周期的变化。将一个典型的阻力变化周期分成三个特征时段,对湍流脉动能的波谱、频谱变化规律以及展向振动壁面抑制壁湍流、实现减阻的内在机理进行了分析。结果表明,总涡能在得到不同程度抑制的同时,三个特征时段相关物理量的谱具有不同的变化特点。从二维能谱能量的分布上看,谱能在整体下降的同时,在波数相对较低的区域,能量有向某一波矢方向集中的趋势。结合近壁湍流拟序结构的变化规律,揭示出展向振动壁面减阻是两种机理共同作用的结果,特定时段由某一机理起主导作用。
2)对展向振动、流向传播的波动壁面的流动控制和减阻问题进行了研究。讨论了流向波动参数(波数)对相关统计量以及壁面阻力的影响。根据诱导边界层(广义Stokes层)调制下近壁湍流拟序结构以及湍流猝发事件的变化规律,对此类波动壁面的湍流控制和减阻机理进行了分析。结果表明,当此类波动壁面被用来调制近壁流动时,固有的湍流流场和诱导流场会各自发生变化。仅低频波对湍流流场具有显著影响,可导致近壁湍流拟序结构以及相应的湍流猝发事件的显著变化；波数的增大对于湍流猝发事件的频率和强度增减的影响并不同步,存在一个最优的波数,在其调制下,固有流场对诱导流场的影响最弱,而诱导流场对固有流场的影响显著,减阻效果最好。此外,给出了最优减阻率随振幅及振动周期的变化曲线,对具有最佳减阻效果的最优控制波数的存在进行了确认。
3)对展向行波电磁力(STW)的湍流控制和减阻机理进行了研究。结果表明,展向行波电磁力对湍流的控制过程实质上是一种由电磁力诱导的特殊流场对壁湍流的调制过程。调制流场由一系列的沿流向伸展并沿展向传播的,距离相等、相互平行的负流向涡组成。该流向涡可以减弱湍流固有的正流向涡和兼并湍流固有的负流向涡。于是,在这种反复周期出现的调制波作用下,湍流流场的流向涡结构和条带数逐渐减少,调制流场也因此逐渐主宰壁面边界层,使其层流化发展,这导致了壁面阻力的下降。优化参数控制下,平均减阻率最高可达30%以上。此外,讨论了展向行波电磁力的波长对控制效果的影响,结果表明,对于长波电磁力,固有流场对诱导流场的影响较弱,而诱导流场可以有效地减少固有流场的“条带—涡”结构,抑制湍流猝发活动,故减阻效果理想；对于短波电磁力,诱导流场作用下,近壁湍流上扬、下扫活动明显增强,故增阻效果显著。
4)研究了展向振荡、流向传播的波动电磁力的流动控制和减阻问题。讨论了流向波数对壁面阻力及相关统计量的影响。从展向速度、近壁拟序结构以及湍流猝发事件的频率和强度等方面对其控制壁湍流、实现壁面减阻的内在机理进行了分析。结果表明,当此类波动电磁力被用来调制近壁流动时,控制效果和展向振动、流向传播的波动壁面有许多相似之处,仅低频波对湍流流场具有显著影响,可导致湍流猝发事件的频率和强度的显著变化；波数的增大对于湍流猝发事件的频率和强度增减的影响并不一致,存在一个最佳的控制波数,在其调制下,减阻效果最好。此外,本文的计算结果显示,波动电磁力的振动参数对最佳控制波数和最优减阻率的影响较为显著。优化参数控制下,展向振荡、流向传播的波动电磁力的能量利用效率要高于振荡电磁力。(此部分内容被国际著名物理类期刊Physics of Fluids审稿专家评为The work is well motivated and the results are interesting. They provide new and relevant information on the effect of an oscillating spanwise Lorentz force on a turbulent channel flow. In particular, the effect of a streamwise wavenumber in the force distribution is studied. Overall, I think the manuscript will merit publication in Phyiscs of Fluids.)
The surface of most mobile devices is inwrapped by turbulent bundary layer. The coherent structures of wall turbulence, charactered by the streaks and longitudinal vortex structures, are responsible for the generation of high turbulent skin-friction drag. When spanwise motions imposed by external forcing excitation are applied to modify the near-wall turbulent flow, the coherent structure of wall turbulence can be disrupted effectively, thereby suppressing the production of turbulence near the wall, leading to significant turbulent skin-friction drag reduction. Several different ways of introducing spanwise flow modifications in turbulent channel flow are investigated by direct numerical simulation (DNS) based on standard Fourier-Chebyshev spectral method. The database of turbulent channel flow is produced by DNS. Control strategies can be classified into two categories. The first class uses wall motions, including spanwise wall oscillation, and streamwise travelling wave induced by spanwise wall oscillation; the second class adopts Lorentz forcing exitation, including spanwise and streamwise travelling waves induced by spanwise oscillating Lorentz force. In the simulations, the incompressible Navier-Stokes (N-S) equations are used as the basic control equations to describe the turbulent channel flow. Different boundary conditions or body-force terms are added to the N-S equations according to the adopted control strategies.
1) Flow control and drag reduction due to spanwise wall oscillation in turbulent channel flow are investigated numerically. The skin-friction drag can be reduced significantly by changing the amplitude and the period of the spanwise wall oscillation. Along with the increase of the mean drag reduction rate, time evolution of skin-friction drag becomes steadier and shows longer periodicity. A type drag reduction periodicity is divided to three typical phases to investigate the energy spectra of turbulent fluctuation kinetic energy and flow physics of turbulent channel flow subject to spanwise wall oscillation. It is found that the suppressions of total vorticity energy for these three phases are different. The distribution of the two-dimensional energy spectra of velocity fluctuation show that turbulent fluctuation kinetic energy decrease significantly, and there seems a trend that energy concentrates to a certain wave vector at low-wave number end. Combining these observations with the different states of the near-wall turbulent structures in the three typical phases, the mechanisms of turbulence suppression and drag reduction via spanwise wall oscillation are further analysed. The results suggest that two kinds of mechanisms of turbulence suppression and drag reduction due to spanwise wall oscillation work by turns, one of which is the incline of streamwise vorticity and the streaks induced by oscillation of the wall, leading to creation of a negative spanwise vorticity in the near-wall region, which is in favor of drag reduction, and the other is the relative displacement of the streamwise vorticity and the streaks over the oscillating wall, which can induce the streaks broad and the streamwise vorticity weak.
2) The control and drag reduction in turbulent channel flow via streamwise travelling wave induced by spanwise-wall oscillation are investigated by DNS. The effects of streamwise wave number on the the associated statistics and the skin-friction drag are discussed. The induced Stokes layer, the near-wall flow structure, as well as the turbulent burst events are analysed. In addition, the mechanisms of turbulence suppression and drag reduction via streamwise traveling wave induced by spanwise-wall oscillation are also discussed. The results suggest that the intrinsic and induced flows are modified each other when the travelling wavy wall is utilized to modulate the near-wall turbulent flow. Only the low-frequency waves have the significant influence on the near-wall turbulent structures which induces the great variations of the the turbulent bursting events which can be detected by VITA detective technique. The variations of the frequency and the intensity of the burst are unsynchronized with wave number, and there exists an optimal wave number, in which the influence of the intrinsic flow on the induced flow is weakest, and on the contrary, the intrinsic turbulent flow is modified strongly by the induced flow, and the largest amount of drag reduction is obtained. Otherwise, the effects of the oscillation parameters on the maximum drag reduction rate are presented, further confirming the existence of the optimal wave number for drag reduction.
3) Mechanism for turbulent control and drag reduction in a channel flow subject to a spanwise travelling wave (STW) via Lorentz forcing is investigated numerically. The results show that the intrinsic flow structures are modified by the induced flow on application of STW control to the turbulent bundary layer. The induced flow is consisted by a distinct set of longitudinal vortices, moving along the spanwisae direction with same spanwise spacing, and paralleling with each other. As travelling with the spanwise Lorentz force, these generated longitudinal vortices can suppress positive random longitudinal vortices and merge negative random longitudinal vortices in the near-wall region of the intrinsic turbulent flow. Then, the intrinsic longitudinal vortex structures and the near-wall pairs of streaks are reduced over time. The induced flow finally dominates the near-wall flow and consequently relaminarizes the flow, leading to a reduction in turbulent drag of more than30%. Otherwise, the influences of the spanwise wavelength in the control were discussed. As the wavelength large enough, the influence of the intrinsic flow on the induced flow is weaker, and on the contrary, the streaks and vortex structures in the intrinsic turbulent flow can be reduced remarkably by the induced flow, and correspondingly, the turbulent bursting events are suppressed efficiently. Consequently, larger amount of drag reduction is obtained. If the wavelength is too short, ejection and sweeping activities of the fluid will be enhanced significantly by the induced flow, which will result in notable drag increase.
4) The control and drag reduction in turbulent channel flow via streamwise travelling wave induced by spanwise oscillatory Lorentz force are investigated numerically by DNS. The effects of streamwise control parameter on the skin-friction drag and associated statistics are discussed. The induced spanwise velocity, near-wall flow structures, as well as the frequency and intensity of the turbulent bursting events are analysed to disclose the mechanisms of turbulence suppression and drag reduction via such wavy Lorentz force. The results are very similar with what have been obtained by the streamwise travelling wave induced by spanwise-wall oscillation. Only the low-frequency waves have the significant influence on the near-wall turbulent structures which induces the great variations of the the turbulent bursting events. The results show that the frequency and intensity vary with wave number in the contradictious tendencies. There is an optimal wave number in flow control, in which the maximum is obtained. The optimal wave number and the maximum drag reduction are also affected by the oscillation parameters of the wavy Lorentz forces. In addition, the results show that there is an efficiency improvement for the proposed method compared with the spanwise oscillating Lorentz force method.(The reviewers of Physics of Fluids appraise that the work is well motivated and the results are interesting. They provide new and relevant information on the effect of an oscillating spanwise Lorentz force on a turbulent channel flow. In particular, the effect of a streamwise wavenumber in the force distribution is studied. Overall, I think the manuscript will merit publication in Phyiscs of Fluids.)
引文
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12. Jung W. J., Mangiavacchi N., Akhavan R. Suppression of turbulence in wall-bounded flows by high-frequency spanwise oscillations. Phys Fluids,1992,4 (8):1605-1607.
13. Berger T. W., Kim J., Lee C., Lim J. Turbulent boundary layer control utilizing the Lorentz force. Physics of Fluids,2000,12 (3):631-649.
14. Laadhari F., Skandaji L., Morel R. Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Physics of Fluids,1994,6 (10):3218-3220.
15. Trujillo S., Bogard D., Ball K. Turbulent boundary layer drag reduction using an oscillating wall. AIAA Paper,1997, AIAA 97-1870.
16. Choi K. S., Graham M. Drag reduction of turbulent pipe flows by circular-wall oscillation. Phys Fluids,1998,10(1):7-9.
17. Choi K. S., Clayton B. R. The mechanism of turbulent drag reduction with wall oscillation. Heat Fluid Flow,2001,22 (1):1-9.
18. Du Y., Karniadakis G. E. Suppresing wall turbulence by means of tranverse traveling wave. Science,2000,288:1230-1234.
19. Du Y., Symeonidis V., Karniadakis G. E. Drag reduction in wall bounded turbulence via transverse travelling wave. J. Fluid Mech.,2002,457:1-34.
20. Rediniotis O. K., Lagoudas D. C., Wilson L. N. Development of a shape-memory alloy actuated biomimetic hydrofoil.2000, AIAA 2000-0522.
21.Mani R., Lagoudas D. C., Rediniotis 0. K. Active skin for turbulent drag reduction. SmartMaterials and Structures.2008,17 (3):035004.
22. Quadrio M., Ricco P. Critical assessment of turbulent drag reduction through spanwise wall oscillations. Journal of Fluid Mechanics,2004,521:251-271.
23. Bogard D. C., Ball K. S., Wassen E. Drag reduction for turbulent layer flows using an oscillating wall. AFOSR Report,2000, No. TTCRL 00-2.
24. Baron A., Quadrio M. Turbulent drag reduction by spanwise wall oscillations. Appl. Sci. Res.1996,55:311-326.
25. Miyake Y., Tsujimoto K., Takahashi M. On the mechanism of drag reduction of near-wall turbulence by wall oscillation. JSME. Series B,1997,40 (4):558-566.
26. Dhanak M. R., Si C. On reduction of turbulent friction through spanwise wall oscillations. Journal of Fluid Mechanics,1999,383:175-195.
27. Moin P., Shih T.H., Driver, D. M., Mansour N. N. Direct numerical simulation of a three-dimensional turbulent boundary layer. Phys. Fluids A2,1990,1846-1853.
28.黄伟希,许春晓,崔桂香,张兆顺壁面展向周期振动的槽道湍流减阻机理的研究.力学学报,2004,36(1)：24-30.
29. Zhao H., Wu J. Z., Luo J. S. Turbulent drag reduction by travelling wave of flexible wall. Fluid Dynamics Research,2004,34 (3):175-198.
30.赵辉.槽道湍流直接数值模拟及其行波壁开环主动控制.北京大学硕士学位论文,2003.
31. Quadrio M., Ricco P., Viotti C. Streamwise-travelling waves of spanwise wall velocity for turbulent drag reduction. J. Fluid Mech,2009,627:161-178.
32. Auteri F., Baron A., Belan M., Campanardi G., Quadrio M. Experimental assessment of drag reduction by traveling waves in a turbulent pipe flow. Phys. Fluids,2010,22: 115103.
33. Quadrio M., Ricco P. The laminar generalized Stokes layer and turbulent drag reduction. J. Fluid Mech.,2011,667:135-157.
34.黄乐萍,范宝春.展向行波状Lorentz力的波数对壁湍流控制的影响.宇航学报,2012,33(3)：305-310.
35. Gailitis A., Lielausis O. On a possibility to reduce the hydrodynamical resistance of a plate in an electrolyte. Applied Magnetohydrodynamics,1961,12:143-146.
36. Henoch C., Stace J. Experimental investigation of a salt water turbulent boundary layer modified by an applied streamwise magnetohydrodynamic body force. Phys. Fluids,1995, 7 (6):1371-1383.
37. Crawford C. H., Karniadakis G. E. Reynolds stress analysis of EMHD-controlled wall turbulence, Part I, streamwise forcing. Phys. Fluids,1997,9 (3):788-806.
38. Lim J., Choi H. Control of streamwise vortices with uniform magnetic fluxes. Phys. Fluids,1998,10 (8):1997-2005.
39. Nosenchuck D. M., Brown G. L. Electromagnetic device and method for boundary layer control. US patent,1992:5320309.
40. Rossi L., Thibault J. P. Investigation of wall normal electromagnetic actuator for seawater flow control. J. Turbulence,2002,3:005.
41. Thibault J. P., Rossi, L. Electromagnetic flow control:characteristic numbers and flow regimes of a wall-normal actuator. J. Phys. D:Appl. Phys.2003,36 (20):2559-2568.
42. Bandyopadhyay P. R., Castano J. M. Micro-tiles for electromagnetic turbulence control in saltwater-preliminary investigations. Proceedings of the Forum on control of transitional and turbulent flows, Fluids Engineering Division Conference, ASME,1996.
43. O'Sullivan P. L. Direct numerical simulation of low Reynolds number turbulent channel flow with EMHD control. Phys. Fluids,1998,10(5):1169-1181.
44. Kim S. J., Lee C. M. Investigation of the flow around a circular cylinder under the influence of an electromagnetic force. Experiments in fluids,2000,28:252-260.
45. Chen Z. H., Aubry N. Active control of cylinder wake. Communications in nonlinear science and numerical simulation,2005,10:205-216.
46.陈耀慧,范宝春,周本谋,陈志华,张辉,李鸿志.翼型绕流的电磁力控制.力学学报,2008,40(1)：121-127.
47. Zhang H., Fan B.C., Chen Z.H., Computations of optimal cylinder flow control in weakly conductive fluids, Computers & fluids,2010,39 (8):1261-1266.
48.张辉,范宝春,陈志华,董刚,周本谋.电磁场中圆柱绕流的开环和优化控制.科学通报,2008,53(9)：1026-1031.
49.蒋小勤,刘巨斌,魏中磊.电磁力对湍流边界层相干结构的影响.水动力学研究与进展,2005,20(4)：458-464.
50. Lee C., Kim J. Control of the viscous sublayer for drag reduction. Physics of Fluids,2002, 14 (7):2523-2529.
51. Lee J. H., Sung H. J. Response of a spatially developing turbulent boundary layer to a spanwise oscillating electromagnetic force. Journal of Turbulence,2005,6 (39):2-15.
52. Pang J., Choi K. S. Aessopos A, Yoshida H. Control of near-wall turbulence for drag reduction by spanwise oscillating Loretz force.2004, AIAA 2004-2117.
53. Pang J., Choi K. S. Turbulent drag reduction by Lorentz force oscillation. Physics of Fluids,2004,16 (5):35-38.
54. Xu P., Choi K. S., Jukes T. Near-wall trubulent flow control for drag reduction by Lorentz force. European drag reduction and flow control meeting, Ischia, Italy,2006.
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