前缘曲率变化对平板边界层感受性问题的影响
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摘要
边界层感受性问题是层流向湍流转捩的初始阶段,是实现边界层转捩预测和控制的关键环节.目前已有的研究成果显示,在声波扰动或涡波扰动作用下前缘曲率变化对边界层感受性机制有着显著的影响.本文采用直接数值模拟方法,研究了在自由来流湍流作用下具有不同椭圆形前缘平板边界层感受性问题,揭示椭圆形前缘曲率变化对平板边界层内被激发出Tollmien-Schlichting (T-S)波波包的感受性机制以及波包向前传播群速度的影响;通过快速傅里叶分析方法从波包中提取获得了不同频率的T-S波,详细分析了前缘曲率变化对不同频率的T-S波的幅值、色散关系、增长率、相速度以及形状函数的作用;确定了前缘曲率在平板边界层内激发T-S波的感受性过程中所占据的地位.通过上述研究能够进一步认识和理解边界层感受性机制,从而丰富和完善了流动稳定性理论.
Boundary-layer receptivity is the initial stage of the laminar-turbulent transition, which is the key step to implement the prediction and control of laminar-turbulent transition in the boundary layer. Current studies show that under the action of acoustic wave or vortical disturbance, the variation of leading-edge curvature significantly affects the boundarylayer receptivity. Additionally, the free-stream turbulence is universal in nature. Therefore, direct numerical simulation is performed in this paper to study the receptivity to free-stream turbulence in the flat-plate boundary layer with an elliptic leading edge. To discretize the Navier-Stokes equation, a modified fourth-order Runge-Kutta scheme is introduced for the temporal discretization; the high-order compact finite difference scheme is utilized for the x-and y-direction spatial discretization; the Fourier transform is conducted in the z-direction. The pressure Helmholtz equation is solved by iterating a fourth-order finite difference scheme. In addition, the Jaccobi transform is introduced to convert the curvilinear coordinate system into Cartesian coordinate system. And elliptic equation technique is adopted to generate the body-fitted mesh. Then the effect of elliptic leading-edge curvature on the receptivity mechanism and the propagation speed of the excited Tollmien-Schlichting(T-S) wave packet in the flat-plate boundary layer are revealed. Subsequently,a group of multi-frequency T-S waves is extracted from the T-S wave packets by temporal fast Fourier transform. The influences of different leading-edge curvatures on the amplitudes, dispersion relations, growth rates, phases and shape functions of the excited T-S waves are analyzed in detail. Finally, the position occupied by leading-edge curvature in the boundary-layer receptivity process for the excitation of T-S wave is also confirmed. The numerical results show that the more intensive receptivity is triggered in the smaller leading-edge curvature; on the contrary, the less intensive receptivity is triggered in the greater leading-edge curvature. But in different leading-edge curvatures, the structures of the excited T-S wave packets are almost identical, and the group velocity is close to constant, which is approximate to one-third of the free-stream velocity. Similarly, the greater amplitude of the excited T-S wave can be induced with the smaller leading-edge curvature; whereas the smaller amplitude of the excited T-S wave can be induced with the greater leading-edge curvature. Moreover, the dispersion relations, growth rates, phases and shape function of the excited T-S waves in the boundary layer are found to be nearly invariable in different leading-edge curvatures. Through the above study, a further step can be made to understand the boundary-layer leading-edge receptivity and also improve the theory of the hydrodynamic stability.
Boundary-layer receptivity is the initial stage of the laminar-turbulent transition, which is the key step to implement the prediction and control of laminar-turbulent transition in the boundary layer. Current studies show that under the action of acoustic wave or vortical disturbance, the variation of leading-edge curvature significantly affects the boundarylayer receptivity. Additionally, the free-stream turbulence is universal in nature. Therefore, direct numerical simulation is performed in this paper to study the receptivity to free-stream turbulence in the flat-plate boundary layer with an elliptic leading edge. To discretize the Navier-Stokes equation, a modified fourth-order Runge-Kutta scheme is introduced for the temporal discretization; the high-order compact finite difference scheme is utilized for the x-and y-direction spatial discretization; the Fourier transform is conducted in the z-direction. The pressure Helmholtz equation is solved by iterating a fourth-order finite difference scheme. In addition, the Jaccobi transform is introduced to convert the curvilinear coordinate system into Cartesian coordinate system. And elliptic equation technique is adopted to generate the body-fitted mesh. Then the effect of elliptic leading-edge curvature on the receptivity mechanism and the propagation speed of the excited Tollmien-Schlichting(T-S) wave packet in the flat-plate boundary layer are revealed. Subsequently,a group of multi-frequency T-S waves is extracted from the T-S wave packets by temporal fast Fourier transform. The influences of different leading-edge curvatures on the amplitudes, dispersion relations, growth rates, phases and shape functions of the excited T-S waves are analyzed in detail. Finally, the position occupied by leading-edge curvature in the boundary-layer receptivity process for the excitation of T-S wave is also confirmed. The numerical results show that the more intensive receptivity is triggered in the smaller leading-edge curvature; on the contrary, the less intensive receptivity is triggered in the greater leading-edge curvature. But in different leading-edge curvatures, the structures of the excited T-S wave packets are almost identical, and the group velocity is close to constant, which is approximate to one-third of the free-stream velocity. Similarly, the greater amplitude of the excited T-S wave can be induced with the smaller leading-edge curvature; whereas the smaller amplitude of the excited T-S wave can be induced with the greater leading-edge curvature. Moreover, the dispersion relations, growth rates, phases and shape function of the excited T-S waves in the boundary layer are found to be nearly invariable in different leading-edge curvatures. Through the above study, a further step can be made to understand the boundary-layer leading-edge receptivity and also improve the theory of the hydrodynamic stability.
引文
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[13]Lu C G,Shen L Y 2016 Acta Phys.Sin.65 194701(in Chinese)[陆昌根,沈露予2016物理学报65 194701]
[14]Hammerton P W,Kerschen E J 1996 J.Fluid Mech.310 243
[15]Hammerton P W,Kerschen E J 1997 J.Fluid Mech.353 205
[16]Lin N,Reed H L,Saric W S 1992 Instability,Transition,and Turbulence(New York:Springer)pp421–440
[17]Fuciarelli D,Reed H,Lyttle I 2000 AIAA J.38 1159
[18]Wanderley J B V,Corke T C 2001 J.Fluid Mech.429 1
[19]Buter T A,Reed H L 1994 Phys.Fluid 6 3368
[20]Schrader L U,Brandt L,Mavriplis C,Henningson D S2010 J.Fluid Mech.653 245
[21]Hoffmann K A,Chiang S T 2000 Computational Fluid Dynamics(Vol.I)(Wichita:Engineering Education System)
[22]Shen L,Lu C 2016 Appl.Math.Mech.37 349
[23]Jacobs R G,Durbin P A 2001 J.Fluid Mech.428 185
[24]Dietz A J 1998 AIAA J.36 1171
[2]Saric W S,Reed H L,Kerschen E J 2002 Annu.Rev.Fluid Mech.34 291
[3]Goldstein M E,Hultgren L S 1989 Annu.Rev.Fluid Mech.21 137
[4]Goldstein M E 1985 J.Fluid Mech.154 509
[5]Ruban A I 1984 Fluid Dynam.19 709
[6]Crouch J D 1992 Phys.Fluid A 4 1408
[7]Choudhari M,Streett C L 1992 Phys.Fluid A 4 2495
[8]Bertolotti F P 1997 Phys.Fluid 9 2286
[9]Goldstein M E 1983 J.Fluid Mech.127 59
[10]Goldstein M E,Sockol P M,Sanz J 1983 J.Fluid Mech.129 443
[11]Goldstein M E,Wundrow D W 1998 Theoret.Comput.Fluid Dyn.10 171
[12]Heinrich R A,Kerschen E J 1989 Z.Angew.Math.Mech.69 T596
[13]Lu C G,Shen L Y 2016 Acta Phys.Sin.65 194701(in Chinese)[陆昌根,沈露予2016物理学报65 194701]
[14]Hammerton P W,Kerschen E J 1996 J.Fluid Mech.310 243
[15]Hammerton P W,Kerschen E J 1997 J.Fluid Mech.353 205
[16]Lin N,Reed H L,Saric W S 1992 Instability,Transition,and Turbulence(New York:Springer)pp421–440
[17]Fuciarelli D,Reed H,Lyttle I 2000 AIAA J.38 1159
[18]Wanderley J B V,Corke T C 2001 J.Fluid Mech.429 1
[19]Buter T A,Reed H L 1994 Phys.Fluid 6 3368
[20]Schrader L U,Brandt L,Mavriplis C,Henningson D S2010 J.Fluid Mech.653 245
[21]Hoffmann K A,Chiang S T 2000 Computational Fluid Dynamics(Vol.I)(Wichita:Engineering Education System)
[22]Shen L,Lu C 2016 Appl.Math.Mech.37 349
[23]Jacobs R G,Durbin P A 2001 J.Fluid Mech.428 185
[24]Dietz A J 1998 AIAA J.36 1171