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Aiming at delaying boundary-layer transition of hypersonic vehicles, the second-mode wave in the boundary layer of a Mach 6 flat plate is studied. Linear stability theory (LST) and direct numerical simulations (DNS) are used to investigate the discrete modes and the relation between the suppressing effect of second-mode wave and the location of transverse rectangular micro-groove (0.4 mm in width), respectively. The LST results show that vortex/entropy waves cause the branch types of Mack’s second mode and “mode I” modes (usually derived from fast acoustic waves) to change. The DNS results show that the influence of the grooved surface on the base flow depends on the streamwise location (or boundary-layer thickness). As the grooved surface shifts backward (or thickness increases), the influence of intensity on the base flow decreases, and the friction resistance coefficient
$ C{d_{\text{f}}} $ , differential pressure resistance coefficient$ C{d_{\text{p}}} $ and total resistance coefficient$ C{d_x} $ of the grooved surface also decrease. It is found that the grooves located in front of the synchronization region of the fast mode and slow mode still have an inhibitory effect on the second-mode wave, which is different from the effect of small-sized (micrometer scale) micro-pores reported in the literature. It is also found that the suppression effect on the second-mode wave is best when the grooves are arranged in the vicinity of the maximum growth-rate point or at the location of the synchronization interval of the fast mode and slow mode.-
Keywords:
- hypersonic /
- linear stability /
- transition control /
- rectangular micro-grooves
[1] Richie G 1999 AIAA 4435
[2] Bertin J J, Cummings R M 2006 Annu. Rev. Fluid Mech 38 129
Google Scholar
[3] Whitehead A 1989 AIAA 5013
[4] Morkovin M V 1994 Bull. Am. Phys. Soc 39 1882
[5] Fedorov A 2011 Annu. Rev. Fluid Mech 43 79
Google Scholar
[6] Malmuth N, Fedorov A, Shalaev V, Cole J, Khokhlov A, Hites M, Williams D 1998 Theoretical Fluid Mechanics Meeting Albuquerque, NM, USA, June 15–18, 1998 p2695
[7] Fedorov A V, Malmuth N D, Rasheed A, Hornung H G 2001 AIAA J 39 605
Google Scholar
[8] Rasheed A, Hornung H G, Fedorov A V, Malmuth N D 2002 AIAA J 40 481
Google Scholar
[9] Chokani N, Bountin D A, Shiplyuk A N, Maslov A A 2005 AIAA J 43 149
Google Scholar
[10] Egorov I V, Fedorov A V, Soudakov V G 2008 J. Fluid Mech 601 165
Google Scholar
[11] Egorov I V, Fedorov A V, Novikov A V, Soudakov V G 2007 AIAA 948
[12] Dong M, Li C 2021 AIAA J 59 2368
Google Scholar
[13] Long T H, Dong Y, Zhao R, Wen Z Y 2021 Phys. Fluids 33 054105
Google Scholar
[14] Brès G A, Colonius T, Fedorov A V 2010 AIAA J 48 267
Google Scholar
[15] Zhao R, Liu T, Wen C Y, Zhu J, Chen L 2018 AIAA J 56 2942
Google Scholar
[16] 涂国华, 陈坚强, 袁先旭, 杨强, 张毅锋 2018 空气动力学报 36 273
Tu G H, Chen J Q, Yuan X X, Yang Q, Zhang Y F 2018 Acta Aerodyn. Sin. 36 273 (in Chinese)
[17] 郭启龙, 涂国华, 陈坚强, 袁先旭, 万兵兵 2020 航空动力学报 35 135
Guo Q L, Tu G H, Chen J Q, Yuan X X, Wan B B 2020 J. Aerosp. Power 35 135 (in Chinese)
[18] Guo Q L, Li C, Tu G H, Chen J J, Wan B B, Liu Y 2021 Asia Conference on Mechanical and Aerospace Engineering, Athens, Greece, July 14–17, 2020 p012053
[19] Wang X W, Zhong X L 2008 AIAA 4382
[20] Wang X W, Zhong X L 2012 Phys. Fluids 24 1441
[21] Lukashevich S V, Morozov S O, Shiplyuk A N 2016 J. Appl. Mech. Tech. Phys 57 873
Google Scholar
[22] Zhao R, Wen C Y, Long T H, Tian X D, Zhou L, Wu Y 2019 AIAA J 57 5061
Google Scholar
[23] 孔维萱, 闫超, 赵瑞 2013 航空学报 34 2249
Kong W X, Yan C, Zhao R 2013 Acta Aeronaut. Astronaut. Sin. 34 2249 (in Chinese)
[24] Brès G A, Inkman M, Colonius T, Fedorov A V 2013 J. Fluid Mech 726 312
Google Scholar
[25] Zhao R, Liu T, Wen C Y, Zhu J, Chen L 2019 Phys. Rev. Appl 11 044015
Google Scholar
[26] Zhao R, Dong Y, Zhang X X, Wen Z Y, Long T H, Wu Y 2021 AIAA J 59 1893
Google Scholar
[27] Ma Y B, Zhong X L 2003 J. Fluid Mech 488 31
Google Scholar
[28] Tumin A, Wang X W, Zhong X L 2011 AIAA J 49 463
Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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图 1 增长率
$ - {\alpha _{\text{i}}}$ 随频率${\omega _{\text{r}}}$ 的变化Figure 1. Growth rate
$ - {\alpha _{\text{i}}}$ versus frequency${\omega _{\text{r}}}$ .
图 2 马赫数6平板边界层离散模态的流向演化; (a)
$ x = 50 $ mm, (b)$ x = 200 $ mm, (c)$ x = 1000 $ mmFigure 2. Streamwise evolution of discrete modes of the Ma6 flat plate boundary layer: (a)
$ x = 50 $ mm, (b)$ x = 200 $ mm, (c)$ x = $ $ 1000 $ mm.
图 3 不同流向位置的离散谱和连续谱分 (a)
$ x = 50 $ mm; (b)$ x = 200 $ mm; (c)$ x = 1000 $ mmFigure 3. Discrete and continuum spectrum at different streamwise locations: (a)
$ x = 50 $ mm; (b)$ x = 200 $ mm; (c)$ x = 1000 $ mm.
图 4 Mack模态的线性稳定性分析 (a) 中性曲线; (b) N值曲线
Figure 4. Linear stability analysis of Mack modes: (a) Neutral curves; (b) N-value curves.
图 5 扰动型函数 (a) 流向速度实部和虚部; (b) 温度实部和虚部
Figure 5. Perturbation shape function: (a) Streamwise velocity real and imaginary parts; (b) temperature real and imaginary parts.
图 6 400 kHz的Mack模态沿流向发展情况 (a) 增长率
$ - {\alpha _{\text{i}}}$ ; (b) 相速度$c$ Figure 6. Development of the 400 kHz Mack mode perturbation along the flow direction: (a) Growth rate
$ - {\alpha _{\text{i}}}$ ; (b) phase velocity$c$ .
图 7 开槽位置
Figure 7. Grooving locations.
图 8 流向140 mm处基本流剖面 (a) 无量纲流向速度; (b) 无量纲温度
Figure 8. Basic flow profile at the 140 mm streamwise location: (a) Dimensionless streamwise velocity; (b) dimensionless temperature.
图 9 施加第二模态扰动后的瞬态压力云图 (a) 标准网格; (b) 加密网格
Figure 9. Transient pressure contours after imposing the second-mode perturbation: (a) Standard grid; (b) fine grids.
图 10 标准网格和加密网格上400 kHz第二模态扰动幅值沿流向演化.
Figure 10. Evolution of the 400 kHz second-mode amplitude in the standard grid and the fine grid.
图 11 基本流压力云图
Figure 11. Pressure contours of the base flow.
图 12 A1—A5工况基本流修正压力剖面分布 (a) 首槽中心线; (b) 尾槽中心线
Figure 12. Pressure correction of the A1–A5 basic flow: (a) First groove centerline; (b) tail groove centerline.
图 13 阻力系数(
$C{d_x}, C{d_{\text{f}}}, C{d_{\text{p}}}$ )随开槽位置$ {x_{\text{l}}} $ 的变化Figure 13. Drag coefficients (
$C{d_x}, C{d_{\text{f}}}, C{d_{\text{p}}}$ ) versus grooving locations$ {x_{\text{l}}} $ .
图 14 脉动压力云图 (a)—(f) A0—A5工况
Figure 14. Pulsating pressure: (a)–(f) A0–A5 cases in turn.
图 15 壁面脉动压力沿流向分布 (a)—(e) A1—A5工况壁面脉动压力
Figure 15. Distributions of the wall pressure fluctuation: (a)–(e) The A1–A5 cases.
图 16 400 kHz第二模态沿流向的发展 (a) A0—A5工况流向[100, 300] mm扰动幅值; (b) A0, A4, A5工况在加长区[300, 400] mm扰动幅值
Figure 16. Development of the 400 kHz second mode along the streamwise direction: (a) A0–A5 cases flow direction[100, 300] mm disturbance amplitude; (b) A0, A4, and A5 cases in the extended area of [300, 400] mm.
图 17 透射系数T 随开槽起始位置
${x_{\text{l}}}$ 的变化Figure 17. Variation of transmission coefficient with grooving location.
表 1 来流参数设置
Table 1. Free stream parameter setting.
马赫数Ma 单位
雷诺数 Re/m壁温 Tw/K 来流
密度
ρ/(kg·m–3)来流
温度 Te/K普朗
特数 Pr比热比 γ 6.0 $1 \times {10^7}$ $300{\text{ }}$ $0.0184{\text{ }}$ 216.65 0.72 1.4 
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表 2 开槽位置参数
Table 2. Grooving location parameters.
Cases 开槽起始位置${x_{\text{l}}}/{L_0}$ 开槽区间 A0 — — A1 120 (120, 136) A2 170 (170, 186) A3 200 (200, 216) A4 245 (245, 261) A5 270 (270, 286)
DownLoad: CSV
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[1] Richie G 1999 AIAA 4435
[2] Bertin J J, Cummings R M 2006 Annu. Rev. Fluid Mech 38 129
Google Scholar
[3] Whitehead A 1989 AIAA 5013
[4] Morkovin M V 1994 Bull. Am. Phys. Soc 39 1882
[5] Fedorov A 2011 Annu. Rev. Fluid Mech 43 79
Google Scholar
[6] Malmuth N, Fedorov A, Shalaev V, Cole J, Khokhlov A, Hites M, Williams D 1998 Theoretical Fluid Mechanics Meeting Albuquerque, NM, USA, June 15–18, 1998 p2695
[7] Fedorov A V, Malmuth N D, Rasheed A, Hornung H G 2001 AIAA J 39 605
Google Scholar
[8] Rasheed A, Hornung H G, Fedorov A V, Malmuth N D 2002 AIAA J 40 481
Google Scholar
[9] Chokani N, Bountin D A, Shiplyuk A N, Maslov A A 2005 AIAA J 43 149
Google Scholar
[10] Egorov I V, Fedorov A V, Soudakov V G 2008 J. Fluid Mech 601 165
Google Scholar
[11] Egorov I V, Fedorov A V, Novikov A V, Soudakov V G 2007 AIAA 948
[12] Dong M, Li C 2021 AIAA J 59 2368
Google Scholar
[13] Long T H, Dong Y, Zhao R, Wen Z Y 2021 Phys. Fluids 33 054105
Google Scholar
[14] Brès G A, Colonius T, Fedorov A V 2010 AIAA J 48 267
Google Scholar
[15] Zhao R, Liu T, Wen C Y, Zhu J, Chen L 2018 AIAA J 56 2942
Google Scholar
[16] 涂国华, 陈坚强, 袁先旭, 杨强, 张毅锋 2018 空气动力学报 36 273
Tu G H, Chen J Q, Yuan X X, Yang Q, Zhang Y F 2018 Acta Aerodyn. Sin. 36 273 (in Chinese)
[17] 郭启龙, 涂国华, 陈坚强, 袁先旭, 万兵兵 2020 航空动力学报 35 135
Guo Q L, Tu G H, Chen J Q, Yuan X X, Wan B B 2020 J. Aerosp. Power 35 135 (in Chinese)
[18] Guo Q L, Li C, Tu G H, Chen J J, Wan B B, Liu Y 2021 Asia Conference on Mechanical and Aerospace Engineering, Athens, Greece, July 14–17, 2020 p012053
[19] Wang X W, Zhong X L 2008 AIAA 4382
[20] Wang X W, Zhong X L 2012 Phys. Fluids 24 1441
[21] Lukashevich S V, Morozov S O, Shiplyuk A N 2016 J. Appl. Mech. Tech. Phys 57 873
Google Scholar
[22] Zhao R, Wen C Y, Long T H, Tian X D, Zhou L, Wu Y 2019 AIAA J 57 5061
Google Scholar
[23] 孔维萱, 闫超, 赵瑞 2013 航空学报 34 2249
Kong W X, Yan C, Zhao R 2013 Acta Aeronaut. Astronaut. Sin. 34 2249 (in Chinese)
[24] Brès G A, Inkman M, Colonius T, Fedorov A V 2013 J. Fluid Mech 726 312
Google Scholar
[25] Zhao R, Liu T, Wen C Y, Zhu J, Chen L 2019 Phys. Rev. Appl 11 044015
Google Scholar
[26] Zhao R, Dong Y, Zhang X X, Wen Z Y, Long T H, Wu Y 2021 AIAA J 59 1893
Google Scholar
[27] Ma Y B, Zhong X L 2003 J. Fluid Mech 488 31
Google Scholar
[28] Tumin A, Wang X W, Zhong X L 2011 AIAA J 49 463
Google Scholar
[29] Liu Y, Guo Q L, Tu G H, Yang Q, Yuan X X, Wan B B 2021 International Conference on Mechanical Engineering and Automation Science, Seoul, South Korea, October 28–30, 2021 p132
[30] Zhang H X, Zhang L P, Zhang S H, Li Q 2017 Comput. Fluids 154 371
Google Scholar
[31] Sandham N D, Lüdeke H 2009 AIAA 1288
[32] Tullio N D, Sandham N D 2010 Phys. Fluids 22 094105
Google Scholar
[33] Zhao R, Wen C Y, Tian X D, Yuan W 2018 Int. J. Heat Mass Transfer 121 986
Google Scholar
[34] Wartemann V, Lüdeke H, Sandham N 2009 AIAA 7202
[35] Fedorov A V, Tumin A 2010 AIAA 5003
[36] Liu Z Y, Yu M 2017 AIAA 2247
[37] Deng X G, Mao M L, Tu G H, Zhang H X, Zhang Y F 2012 Commun. Comput. Phys 11 1081
Google Scholar
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