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Three-dimensional boundary-layer receptivity is the first stage of the laminar-turbulent transition in a three-dimensional boundary layer, and also a key issue for predicting and controlling the laminar-turbulent transition in the three-dimensional boundary layer. At a high turbulence level, the three-dimensional boundary-layer instability in the transition is caused mainly by the unsteady cross-flow vortices. And the leading-edge curvature has a significant influence on three-dimensional boundary-layer receptivity. In view of this, the direct numerical simulation is utilized in this paper to study the mechanism of receptivity to exciting unsteady cross-flow vortices in the three-dimensional (swept-plate) boundary layer with various elliptic leading edges. In order to solve the Navier-Stokes equation numerically, a modified fourth-order Runge-Kutta scheme is introduced for discretization in time; high-order compact finite difference schemes are utilized for discretization in the x-and y-direction; and Fourier transform is used in the z-direction. The pressure Helmholtz equation is solved by a fourth-order iterative scheme. Additionally, the numerical calculation is performed in the curvilinear coordinate system via Jaccobi transform. And the elliptic equation technique is used to gene-rate the body-fitted mesh. The effect of leading-edge curvature on the propagation speed and direction, distribution and receptivity coefficient of the excited unsteady cross-flow vortex wave packet, and the amplitude, dispersion relation and growth rate of the extracted unsteady cross-flow vortex are revealed. In addition, the inner link among the receptivity to unsteady cross-flow vortex, intensity, and direction of free-stream turbulence is established. Furthermore, the receptivity to anisotropic free-stream turbulence is also analyzed in detail. The numerical results indicate that the more intense receptivity to the unsteady cross-flow vortex wave packets is triggered with a smaller leading-edge curvature; whereas, the less intense receptivity is triggered with a greater leading-edge curvature. The receptivity to the unsteady cross-flow vortex wave packets in different curvatures are also found to vary with the angle of free-stream turbulence. Moreover, the anisotropic degree of free-stream turbulence can affect the excitation of the unsteady cross-flow vortex obviously. Through the above study, a further step can be taken to understand the prediction and control of laminar-turbulent transition in the three-dimensional boundary layer and also improve the theory of the hydrodynamic stability.
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Keywords:
- receptivity /
- leading-edge curvature /
- unsteady cross-flow vortex
[1] Saric W S, Reed H L, Edward W B 2003 Annu. Rev. Fluid. Mech. 35 413
[2] Bippes H 1999 Prog. Aerosp. Sci. 35 363
[3] Radeztsky R H, Reibert M S, Saric W S 1999 AIAA J. 37 1370
[4] Deyhle H, Bippes H 1996 J. Fluid. Mech. 316 73
[5] Reibert M S, Saric W S 1997 28th Fluid Dynamics Conference Snowmass Village, CO, USA, June 29-July 2, 1997 p1816
[6] Kurz H B E, Kloker M J 2014 J. Fluid. Mech. 755 62
[7] Bertolotti F P 2000 Phys. Fluid. 12 1799
[8] Collis S S, Lele S K 1999 J. Fluid. Mech. 380 141
[9] Schrader L U, Brandt L, Henningson D S 2009 J. Fluid. Mech. 618 209
[10] Schrader L U, Brandt L, Mavriplis C 2010 J. Fluid. Mech. 653 245
[11] Tempelmann D, Schrader L U, Hanifi A, et al. 2011 6th AIAA Theoretical Fluid Mechanics Conference Honolulu, Hawaii, USA, June 27-30, 2011 p3294
[12] Tempelmann D, Schrader L U, Hanifi A 2012 J. Fluid. Mech. 711 516
[13] Borodulin V I, Ivanov A V, Kachanov Y S 2013 J. Fluid. Mech. 716 487
[14] Shen L Y, Lu C G 2016 Acta Phys. Sin. 65 014703 (in Chinese)[沈露予, 陆昌根 2016 物理学报 65 014703]
[15] Lu C G, Shen L Y 2017 Acta Phys. Sin. 66 204702 (in Chinese)[陆昌根, 沈露予 2017 物理学报 66 204702]
[16] Lin R S, Malik M R 1997 J. Fluid. Mech. 333 125
[17] Shen L Y, Lu C G 2018 Acta Phys. Sin. 67 184703 (in Chinese)[沈露予, 陆昌根 2018 物理学报 67 184703]
[18] Hoffmann K A, Chiang S T 2000 Computational Fluid Dynamics (Vol. I) (Kansas: Engineering Education System) p358
[19] Shen L Y, Lu C G 2016 Appl. Math. Mech. 37 349
[20] Zhang Y, Zaki T, Sherwin S, Wu X 2011 6th AIAA Theortical Fluid Mechanics Conference Hawaii, USA, June 27-30, 2011 p3292
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[1] Saric W S, Reed H L, Edward W B 2003 Annu. Rev. Fluid. Mech. 35 413
[2] Bippes H 1999 Prog. Aerosp. Sci. 35 363
[3] Radeztsky R H, Reibert M S, Saric W S 1999 AIAA J. 37 1370
[4] Deyhle H, Bippes H 1996 J. Fluid. Mech. 316 73
[5] Reibert M S, Saric W S 1997 28th Fluid Dynamics Conference Snowmass Village, CO, USA, June 29-July 2, 1997 p1816
[6] Kurz H B E, Kloker M J 2014 J. Fluid. Mech. 755 62
[7] Bertolotti F P 2000 Phys. Fluid. 12 1799
[8] Collis S S, Lele S K 1999 J. Fluid. Mech. 380 141
[9] Schrader L U, Brandt L, Henningson D S 2009 J. Fluid. Mech. 618 209
[10] Schrader L U, Brandt L, Mavriplis C 2010 J. Fluid. Mech. 653 245
[11] Tempelmann D, Schrader L U, Hanifi A, et al. 2011 6th AIAA Theoretical Fluid Mechanics Conference Honolulu, Hawaii, USA, June 27-30, 2011 p3294
[12] Tempelmann D, Schrader L U, Hanifi A 2012 J. Fluid. Mech. 711 516
[13] Borodulin V I, Ivanov A V, Kachanov Y S 2013 J. Fluid. Mech. 716 487
[14] Shen L Y, Lu C G 2016 Acta Phys. Sin. 65 014703 (in Chinese)[沈露予, 陆昌根 2016 物理学报 65 014703]
[15] Lu C G, Shen L Y 2017 Acta Phys. Sin. 66 204702 (in Chinese)[陆昌根, 沈露予 2017 物理学报 66 204702]
[16] Lin R S, Malik M R 1997 J. Fluid. Mech. 333 125
[17] Shen L Y, Lu C G 2018 Acta Phys. Sin. 67 184703 (in Chinese)[沈露予, 陆昌根 2018 物理学报 67 184703]
[18] Hoffmann K A, Chiang S T 2000 Computational Fluid Dynamics (Vol. I) (Kansas: Engineering Education System) p358
[19] Shen L Y, Lu C G 2016 Appl. Math. Mech. 37 349
[20] Zhang Y, Zaki T, Sherwin S, Wu X 2011 6th AIAA Theortical Fluid Mechanics Conference Hawaii, USA, June 27-30, 2011 p3292
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