Application of power law to vertical distribution of longshore currents
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摘要
The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power law is not commonly used to describe the profile of longshore currents. In this paper, however, a power-type formula is proposed to describe the vertical profiles of longshore currents. The formula has two parameters: the power law index(a) and the depth-averaged velocity. Based on previous studies, power law indices were set as a = 1/10 and a = 1/7. Depth-averaged velocity can be obtained through measurement. The fitting of the measured velocity profiles to a = 1/10 and a = 1/7 was assessed for the vertical longshore profiles. The vertical profile of longshore currents is well described by the power-type formula with a = 1/10 for a plane beach. However, for a barred beach, different values of a needed to be used for different regions. For the region from the bar trough to the offshore side of the bar crest, the vertical profiles of longshore currents given by the power-type formula with a = 1/10 and a = 1/7 fit the data well. However, the fit was slightly better with a = 1/10 than that with a = 1/7. For the data over the trough region of cross-shore distribution of the depth-averaged longshore currents, the power formula with a = 1/3 provided a good fit. The formulas with a = 1/10 and a = 1/7 were further examined using published data from four sources covering laboratory and field experiments. The results indicate that the power-type formula fits the data well for the laboratory and field data with a = 1/10.
The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power law is not commonly used to describe the profile of longshore currents. In this paper, however, a power-type formula is proposed to describe the vertical profiles of longshore currents. The formula has two parameters: the power law index(a) and the depth-averaged velocity. Based on previous studies, power law indices were set as a = 1/10 and a = 1/7. Depth-averaged velocity can be obtained through measurement. The fitting of the measured velocity profiles to a = 1/10 and a = 1/7 was assessed for the vertical longshore profiles. The vertical profile of longshore currents is well described by the power-type formula with a = 1/10 for a plane beach. However, for a barred beach, different values of a needed to be used for different regions. For the region from the bar trough to the offshore side of the bar crest, the vertical profiles of longshore currents given by the power-type formula with a = 1/10 and a = 1/7 fit the data well. However, the fit was slightly better with a = 1/10 than that with a = 1/7. For the data over the trough region of cross-shore distribution of the depth-averaged longshore currents, the power formula with a = 1/3 provided a good fit. The formulas with a = 1/10 and a = 1/7 were further examined using published data from four sources covering laboratory and field experiments. The results indicate that the power-type formula fits the data well for the laboratory and field data with a = 1/10.
The vertical profiles of longshore currents have been examined experimentally over plane and barred beaches. In most cases, the vertical profiles of longshore currents are expressed by the logarithmic law. The power law is not commonly used to describe the profile of longshore currents. In this paper, however, a power-type formula is proposed to describe the vertical profiles of longshore currents. The formula has two parameters: the power law index(a) and the depth-averaged velocity. Based on previous studies, power law indices were set as a = 1/10 and a = 1/7. Depth-averaged velocity can be obtained through measurement. The fitting of the measured velocity profiles to a = 1/10 and a = 1/7 was assessed for the vertical longshore profiles. The vertical profile of longshore currents is well described by the power-type formula with a = 1/10 for a plane beach. However, for a barred beach, different values of a needed to be used for different regions. For the region from the bar trough to the offshore side of the bar crest, the vertical profiles of longshore currents given by the power-type formula with a = 1/10 and a = 1/7 fit the data well. However, the fit was slightly better with a = 1/10 than that with a = 1/7. For the data over the trough region of cross-shore distribution of the depth-averaged longshore currents, the power formula with a = 1/3 provided a good fit. The formulas with a = 1/10 and a = 1/7 were further examined using published data from four sources covering laboratory and field experiments. The results indicate that the power-type formula fits the data well for the laboratory and field data with a = 1/10.
引文
Barenblatt, G.I., 2014. Flow, Deformation and Fracture:Lectures on Fluid Mechanics and the Mechanics of Deformable Solids for Mathematicians and Physicists. Cambridge University Press, New York.
Basco, D.R., 1983. Surfzone currents. Coast. Eng. 7(4), 331-355. https://doi.org/10.1016/0378-3839(83)90003-0.
Cheng, N.S., 2007. Power-law index for velocity profiles in open channel flows. Adv. Water Resour. 30(8), 1775-1784. https://doi.org/10.1016/j.advwatres.2007.02.001.
Church, J.C., Thornton, E.B., 1993. Effects of breaking wave induced turbulence within a longshore current model. Coast. Eng. 20(1-2), 1-28.https://doi.org/10.1016/0378-3839(93)90053-B.
Dyer, K.R., 1971. Current velocity profiles in a tidal channel. Geophys. J. R.Astron. Soc. 22(2), 153-161. https://doi.org/10.1111/j.1365-246X.1971.tb03589.x.
Faria, A.F.G., Thornton, E.B., Stanton, T.P., Soares, C.V., 1998. Vertical profiles of longshore currents and related bed shear stress and bottom roughness. J. Geophys. Res. 103(C2), 3217-3232. https://doi.org/10.1029/97JC02265.
Fredsoe, J., Deigaard, R., 1992. Mechanics of Coastal Sediment Transport.World Scientific, Singapore. https://doi.org/10.1142/1546.
Hamilton, D.G., Ebersole, B.A., 2001. Establishing uniform longshore currents in a large-scale sediment transport facility. Coast. Eng. 42(3),199-218. https://doi.org/10.1016/S0378-3839(00)00059-4.
Ren, C.P., Zou, Z.L., Qiu, D.H., 2012. Experimental study of the instabilities of alongshore currents on plane beaches. Coast. Eng. 59(1), 72-89. https://doi.org/10.1016/j.coastaleng.2011.07.004.
Reniers,A.J.H.M., Battjes, J.A., 1997. A laboratory study of longshore currents over barred and non-barred beaches. Coast. Eng. 30(1-2), 1-21.https://doi.org/10.1016/S0378-3839(96)00033-6.
Shen, L.D., Zou, Z.L., Zhao, X.Z., 2018. Linear analysis of longshore currents instability over mild slopes. China Ocean Eng. 32(6), 675-682. https://doi.org/10.1007/s13344-018-0069-y.
Sleath, J.F.A., 1987. Turbulent oscillatory flow over rough beds. J. Fluid Mech.182. 369-409. https://doi.org/10.1017/S0022112087002374.
Visser, P.J., 1991. Laboratory measurements of uniform longshore currents.Coast. Eng. 15(5-6), 563-593. https://doi.org/10.1016/0378-3839(91)90028-F.
Wang, P., Ebersole, B.A., Smith, E.R., Johnson, B.D., 2002. Temporal and spatial variations of surf-zone currents and suspended sedimentconcentration. Coast. Eng. 46(3), 175-211. https://doi.org/10.1016/S0378-3839(02)00091-1.
Wang, Y., Zou, Z.L., 2015. An experimental and numerical study of bimodal velocity profile of longshore currents over mild-slope barred beaches. Ocean Eng. 106, 415-423. https://doi.org/10.1016/j.oceaneng.2015.06.038.
Zhang, Z., Song, Z.Y., Zhang, D., Guo, F., Hu, D., 2016. Application of power law to the currents in a tidal channel. J. Oceanogr. 72(4), 589-600. https://doi.org/10.1007/s10872-016-0353-5.
Zhang, Z.W., Zou, Z.L., 2012. Vertical distribution of longshore currents over plane and barred beaches. J. Hydrodyn. 24(5), 718-728. https://doi.org/10.1016/S1001-6058(11)60296-5.
Basco, D.R., 1983. Surfzone currents. Coast. Eng. 7(4), 331-355. https://doi.org/10.1016/0378-3839(83)90003-0.
Cheng, N.S., 2007. Power-law index for velocity profiles in open channel flows. Adv. Water Resour. 30(8), 1775-1784. https://doi.org/10.1016/j.advwatres.2007.02.001.
Church, J.C., Thornton, E.B., 1993. Effects of breaking wave induced turbulence within a longshore current model. Coast. Eng. 20(1-2), 1-28.https://doi.org/10.1016/0378-3839(93)90053-B.
Dyer, K.R., 1971. Current velocity profiles in a tidal channel. Geophys. J. R.Astron. Soc. 22(2), 153-161. https://doi.org/10.1111/j.1365-246X.1971.tb03589.x.
Faria, A.F.G., Thornton, E.B., Stanton, T.P., Soares, C.V., 1998. Vertical profiles of longshore currents and related bed shear stress and bottom roughness. J. Geophys. Res. 103(C2), 3217-3232. https://doi.org/10.1029/97JC02265.
Fredsoe, J., Deigaard, R., 1992. Mechanics of Coastal Sediment Transport.World Scientific, Singapore. https://doi.org/10.1142/1546.
Hamilton, D.G., Ebersole, B.A., 2001. Establishing uniform longshore currents in a large-scale sediment transport facility. Coast. Eng. 42(3),199-218. https://doi.org/10.1016/S0378-3839(00)00059-4.
Ren, C.P., Zou, Z.L., Qiu, D.H., 2012. Experimental study of the instabilities of alongshore currents on plane beaches. Coast. Eng. 59(1), 72-89. https://doi.org/10.1016/j.coastaleng.2011.07.004.
Reniers,A.J.H.M., Battjes, J.A., 1997. A laboratory study of longshore currents over barred and non-barred beaches. Coast. Eng. 30(1-2), 1-21.https://doi.org/10.1016/S0378-3839(96)00033-6.
Shen, L.D., Zou, Z.L., Zhao, X.Z., 2018. Linear analysis of longshore currents instability over mild slopes. China Ocean Eng. 32(6), 675-682. https://doi.org/10.1007/s13344-018-0069-y.
Sleath, J.F.A., 1987. Turbulent oscillatory flow over rough beds. J. Fluid Mech.182. 369-409. https://doi.org/10.1017/S0022112087002374.
Visser, P.J., 1991. Laboratory measurements of uniform longshore currents.Coast. Eng. 15(5-6), 563-593. https://doi.org/10.1016/0378-3839(91)90028-F.
Wang, P., Ebersole, B.A., Smith, E.R., Johnson, B.D., 2002. Temporal and spatial variations of surf-zone currents and suspended sedimentconcentration. Coast. Eng. 46(3), 175-211. https://doi.org/10.1016/S0378-3839(02)00091-1.
Wang, Y., Zou, Z.L., 2015. An experimental and numerical study of bimodal velocity profile of longshore currents over mild-slope barred beaches. Ocean Eng. 106, 415-423. https://doi.org/10.1016/j.oceaneng.2015.06.038.
Zhang, Z., Song, Z.Y., Zhang, D., Guo, F., Hu, D., 2016. Application of power law to the currents in a tidal channel. J. Oceanogr. 72(4), 589-600. https://doi.org/10.1007/s10872-016-0353-5.
Zhang, Z.W., Zou, Z.L., 2012. Vertical distribution of longshore currents over plane and barred beaches. J. Hydrodyn. 24(5), 718-728. https://doi.org/10.1016/S1001-6058(11)60296-5.