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In this paper, we perform experiments on the time-space characteristics of internal waves generated by horizontally towed bodies with three aspect ratios in a stratified fluid with a halocline. By the real-time measurements of conductivity probe arrays which are arranged symmetrically in the transverse section of the stratified fluid tank, it is shown that the transition between the body-generated internal wave and the wake-generated internal wave is related to a critical Froude number Frc, which is linearly dependent on the aspect ratio. For FrFrc, the correlation velocities of internal waves are consistent with the towing speeds of the towed bodies, indicating that such internal waves in this range are dominated by the body-forced effect. The heights of such body-generated internal waves first increase with the increase of Fr until Fr reaches a certain value of Frp, which is also linearly dependent on the aspect ratio, and then decrease. For FrFrc, the correlation velocities of internal waves are noticeably lower than the towing speeds, indicating that such internal waves in this range are dominated by the wake-forced effect, and that the Froude numbers with respect to the correlation velocities of such internal waves vary in a range from 0.43 to 1.18. The heights of such wake-generated internal waves nearly linearly increase with Fr increasing regardless of the aspect ratio. Moreover, the patterns of body-generated waves are symmetric, while the patterns of wake-generated waves are not symmetric. Based on the experimental results and the equivalent source method which has been proposed to simulate the internal waves generated by a towed sphere, a new equivalent source method is developed to calculate the internal waves generated by towed slender bodies. For the body-generated waves, the method of designing the speed, length and diameter of the equivalent source is proposed. The symmetrical and anti-symmetrical equivalent source and their speed and size are also proposed for the wake-generated waves. The numerical results are in good accordance with the experimental results in the heights and patterns of waves, indicating that such a theoretical method and its parameter settings are reasonable and effective.
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Keywords:
- stratified fluid /
- internal waves /
- equivalent sources /
- towed slender body
[1] Liang J J, Du T, Huang W G, Zeng K, He M X 2016 J. Ship Mech. 20 635 (in Chinese) [梁建军, 杜涛, 黄韦艮, 曾侃, 贺明霞 2016 船舶力学 20 635]
[2] Robey H F 1997 Phys. Fluids 9 3353
[3] Hopfinger E J, Flor J B, Chomaz J M, Bonneton P 1991 Exp. Fluids 11 255
[4] Lin Q, Boyer D L, Fernando H J S 1993 Exp. Fluids 15 147
[5] Chomaz J M, Bonneton P, Hopfinger E J 1993 J. Fluid Mech. 254 1
[6] Bonneton P, Chomaz J M, Hopfinger E J 1993 J. Fluid Mech. 254 23
[7] Wei G, Zhao X Q, Su X B, You Y X 2009 Sci. China: Series G 39 1338 (in Chinese) [魏岗, 赵先奇, 苏晓冰, 尤云祥 2009 中国科学G 39 1338]
[8] Zhao X Q, You Y X, Chen K, Hu T Q, Wei G 2009 J. Shanghai Jiao Tong Univ. 43 1298 (in Chinese) [赵先奇, 尤云祥, 陈科, 胡天群, 魏岗 2009 上海交通大学学报 43 1298]
[9] Wang J, You Y X, Hu T Q, Wang X Q, Zhu M H 2012 Acta Phys. Sin. 61 074701 (in Chinese) [王进, 尤云祥, 胡天群, 王小青, 朱敏慧 2012 61 074701]
[10] Wang J, You Y X, Hu T Q, Zhu M H, Wang X Q, Wei G 2012 Chin. Sci. Bull. 57 606 (in Chinese) [王进, 尤云祥, 胡天群, 朱敏慧, 王小青, 魏岗 2012 科学通报 57 606]
[11] Wang H W, Chen K, You Y X, Zhang X S 2017 Chin. Sci. Bull. 62 2132 (in Chinese) [王宏伟, 陈科, 尤云祥, 张新曙 2017 科学通报 62 2132]
[12] Lighthill J 1978 Waves in Fluid (Cambridge: Cambridge University Press) pp23-30
[13] Keller J B, Munk W H 1970 Phys. Fluids 13 1425
[14] Miles J W 1971 Geo. Fluid Dynamics 2 63
[15] Gray E P 1983 Phys. Fluids 26 2919
[16] Voisin B 1994 J. Fluid Mech. 261 333
[17] Yeung R W, Nguyen T C 1999 J. Eng. Math. 35 85
[18] Broutman D, Rottman J, Eckermann S D 2004 Annu. Rev. Fluid Mech. 36 233
[19] Milder M 1974 Internal Waves Radiated by a Moving Source Technical Report (Vol. 1) (Santa Monica: Defense Advanced Research Projects Agency) pp19-25
[20] You Y X, Zhao X Q, Chen K, Wei G 2009 Acta Phys. Sin. 58 6750 (in Chinese) [尤云祥, 赵先奇, 陈科, 魏岗 2009 58 6750]
[21] Brandt A, Rottier J R 2015 J. Fluid Mech. 769 103
[22] Lin J T, Pao Y H 1979 Annu. Rev. Fluid Mech. 11 317
[23] Gilreath H E, Brandt A 1985 AIAA J. 23 693
[24] Wei G, Wu N, Xu X H, Su X B, You Y X 2011 Acta Phys. Sin. 60 044704 (in Chinese) [魏岗, 吴宁, 徐小辉, 苏晓冰, 尤云祥 2011 60 044704]
[25] Dupont P, Kadri Y, Chomaz J M 2001 Phys. Fluids 13 3223
[26] Druzhinin O A, Papko V V, Sergeev D A, Troitskaya Y I 2006 Izv. Atmos. Ocean. Phys. 42 615
[27] Druzhinin O A 2009 Fluid Dyn. 44 213
[28] Vasholz D P 2011 Theoretical and Computational Fluid Dynamics 25 357
[29] Diamessis P J, Gurka R, Liberzon A 2010 Phys. Fluids 22 086601
[30] Abdilghanie A M, Diamessis P J 2013 J. Fluid Mech. 720 104
[31] Yao Z C, Zhao F, Liang C, Hong F W, Zhang J 2017 J. Ship Mech. 21 8 (in Chinese) [姚志崇, 赵峰, 梁川, 洪方文, 张军 2017 船舶力学 21 8]
[32] Dupont P, Voisin B 1996 Dynamics Atmo. Oceans 23 289
[33] Liang C, Hong F W, Yao Z C 2015 J. Hydrodynamics Ser. A 30 9 (in Chinese) [梁川, 洪方文, 姚志崇 2015 水动力学研究与进展 30 9]
[34] Cai S, Xie J, Xu J, Wang D, Chen Z, Deng X, Long X 2014 Deep Sea Res. Part I 84 73
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[1] Liang J J, Du T, Huang W G, Zeng K, He M X 2016 J. Ship Mech. 20 635 (in Chinese) [梁建军, 杜涛, 黄韦艮, 曾侃, 贺明霞 2016 船舶力学 20 635]
[2] Robey H F 1997 Phys. Fluids 9 3353
[3] Hopfinger E J, Flor J B, Chomaz J M, Bonneton P 1991 Exp. Fluids 11 255
[4] Lin Q, Boyer D L, Fernando H J S 1993 Exp. Fluids 15 147
[5] Chomaz J M, Bonneton P, Hopfinger E J 1993 J. Fluid Mech. 254 1
[6] Bonneton P, Chomaz J M, Hopfinger E J 1993 J. Fluid Mech. 254 23
[7] Wei G, Zhao X Q, Su X B, You Y X 2009 Sci. China: Series G 39 1338 (in Chinese) [魏岗, 赵先奇, 苏晓冰, 尤云祥 2009 中国科学G 39 1338]
[8] Zhao X Q, You Y X, Chen K, Hu T Q, Wei G 2009 J. Shanghai Jiao Tong Univ. 43 1298 (in Chinese) [赵先奇, 尤云祥, 陈科, 胡天群, 魏岗 2009 上海交通大学学报 43 1298]
[9] Wang J, You Y X, Hu T Q, Wang X Q, Zhu M H 2012 Acta Phys. Sin. 61 074701 (in Chinese) [王进, 尤云祥, 胡天群, 王小青, 朱敏慧 2012 61 074701]
[10] Wang J, You Y X, Hu T Q, Zhu M H, Wang X Q, Wei G 2012 Chin. Sci. Bull. 57 606 (in Chinese) [王进, 尤云祥, 胡天群, 朱敏慧, 王小青, 魏岗 2012 科学通报 57 606]
[11] Wang H W, Chen K, You Y X, Zhang X S 2017 Chin. Sci. Bull. 62 2132 (in Chinese) [王宏伟, 陈科, 尤云祥, 张新曙 2017 科学通报 62 2132]
[12] Lighthill J 1978 Waves in Fluid (Cambridge: Cambridge University Press) pp23-30
[13] Keller J B, Munk W H 1970 Phys. Fluids 13 1425
[14] Miles J W 1971 Geo. Fluid Dynamics 2 63
[15] Gray E P 1983 Phys. Fluids 26 2919
[16] Voisin B 1994 J. Fluid Mech. 261 333
[17] Yeung R W, Nguyen T C 1999 J. Eng. Math. 35 85
[18] Broutman D, Rottman J, Eckermann S D 2004 Annu. Rev. Fluid Mech. 36 233
[19] Milder M 1974 Internal Waves Radiated by a Moving Source Technical Report (Vol. 1) (Santa Monica: Defense Advanced Research Projects Agency) pp19-25
[20] You Y X, Zhao X Q, Chen K, Wei G 2009 Acta Phys. Sin. 58 6750 (in Chinese) [尤云祥, 赵先奇, 陈科, 魏岗 2009 58 6750]
[21] Brandt A, Rottier J R 2015 J. Fluid Mech. 769 103
[22] Lin J T, Pao Y H 1979 Annu. Rev. Fluid Mech. 11 317
[23] Gilreath H E, Brandt A 1985 AIAA J. 23 693
[24] Wei G, Wu N, Xu X H, Su X B, You Y X 2011 Acta Phys. Sin. 60 044704 (in Chinese) [魏岗, 吴宁, 徐小辉, 苏晓冰, 尤云祥 2011 60 044704]
[25] Dupont P, Kadri Y, Chomaz J M 2001 Phys. Fluids 13 3223
[26] Druzhinin O A, Papko V V, Sergeev D A, Troitskaya Y I 2006 Izv. Atmos. Ocean. Phys. 42 615
[27] Druzhinin O A 2009 Fluid Dyn. 44 213
[28] Vasholz D P 2011 Theoretical and Computational Fluid Dynamics 25 357
[29] Diamessis P J, Gurka R, Liberzon A 2010 Phys. Fluids 22 086601
[30] Abdilghanie A M, Diamessis P J 2013 J. Fluid Mech. 720 104
[31] Yao Z C, Zhao F, Liang C, Hong F W, Zhang J 2017 J. Ship Mech. 21 8 (in Chinese) [姚志崇, 赵峰, 梁川, 洪方文, 张军 2017 船舶力学 21 8]
[32] Dupont P, Voisin B 1996 Dynamics Atmo. Oceans 23 289
[33] Liang C, Hong F W, Yao Z C 2015 J. Hydrodynamics Ser. A 30 9 (in Chinese) [梁川, 洪方文, 姚志崇 2015 水动力学研究与进展 30 9]
[34] Cai S, Xie J, Xu J, Wang D, Chen Z, Deng X, Long X 2014 Deep Sea Res. Part I 84 73
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