Applied Mathematics and Mechanics >
Analytical solutions of turbulent boundary layer beneath forward-leaning waves
Received date: 2023-07-25
Online published: 2024-04-08
Supported by
the National Key R&D Program of China(2022YFC3204303);the National Natural Science Foundation of China(12202503);the National Natural Science Foundation of China(12132018);the National Natural Science Foundation of China(52394254);Project supported by the National Key R&D Program of China (No.2022YFC3204303) and the National Natural Science Foundation of China (Nos.12202503, 12132018, and 52394254)
Copyright
As a typical nonlinear wave, forward-leaning waves can be frequently encountered in the near-shore areas, which can impact coastal sediment transport significantly. Hence, it is of significance to describe the characteristics of the boundary layer beneath forward-leaning waves accurately, especially for the turbulent boundary layer. In this work, the linearized turbulent boundary layer model with a linear turbulent viscosity coefficient is applied, and the novel expression of the near-bed orbital velocity that has been worked out by the authors for forward-leaning waves of arbitrary forward-leaning degrees is further used to specify the free stream boundary condition of the bottom boundary layer. Then, a variable transformation is found so as to make the equation of the turbulent boundary layer model be solved analytically through a modified Bessel function. Consequently, an explicit analytical solution of the turbulent boundary layer beneath forward-leaning waves is derived by means of variable separation and variable transformation. The analytical solutions of the velocity profile and bottom shear stress of the turbulent boundary layer beneath forward-leaning waves are verified by comparing the present analytical results with typical experimental data available in the previous literature.
Yiqin XIE, Jifu ZHOU, Xu WANG, Jinlong DUAN, Yongjun LU, Shouqian LI . Analytical solutions of turbulent boundary layer beneath forward-leaning waves[J]. Applied Mathematics and Mechanics, 2024 , 45(4) : 695 -710 . DOI: 10.1007/s10483-024-3099-8
| 1 | DEAN, R. G. and DALRYMPLE, R. A. Water Wave Mechanics for Engineers and Scientists, World Scientific Publishing Company, Singapore (1991) |
| 2 | LANDAU, L. D., and LIFSHITZ, E. M. Fluid Mechanics, 2nd ed Beijing World Publishing Corporation, Beijing (1987) |
| 3 | LI, Y. J., CHEN, J. B., ZHOU, J. F., and ZHANG, Q. Large eddy simulation of boundary layer flow under cnoidal waves. Acta Mechanica Sinica, 32 (1), 22- 37 (2016) |
| 4 | STOKES, G. G. On the Theory of Oscillatory Waves, Cambridge University Press, London 314- 326 (1880) |
| 5 | JONSSON, I. G., and CARLSEN, N. A. Experimental and theoretical investigations in an oscillatory turbulent boundary layer. Journal of Hydraulic Research, 14 (1), 45- 60 (1976) |
| 6 | HINO, M., KASHIWAYANAGI, M., NAKAYAMA, A., and HARA, T. Experiments on the turbulence statistics and the structure of a reciprocating oscillatory flow. Journal of Fluid Mechanics, 131, 363- 400 (1983) |
| 7 | SLEATH, J. F. A. Turbulent oscillatory flow over rough beds. Journal of Fluid Mechanics, 182, 369- 409 (1987) |
| 8 | JENSEN, B. L., SUMER, B. M., and FREDSØE, J. Turbulent oscillatory boundary layer at high Reynolds numbers. Journal of Fluid Mechanics, 206, 265- 297 (1989) |
| 9 | LAADHARI, F., SKANDAJI, L., and MORELA, R. Turbulence reduction in a boundary layer by a local spanwise oscillating surface. Physics of Fluids, 6, 3218- 3220 (1994) |
| 10 | VERZICCO, R., and VITTORI, G. Direct simulation of transition in Stokes boundary layer. Physics of Fluids, 8, 1341- 1343 (1996) |
| 11 | CARSTENSEN, S., SUMER, B. M., and FREDSOE, J. A note on turbulent spots over a rough bed in wave boundary layers. Physics of Fluids, 24, 115104 (2012) |
| 12 | KEULEGAN, G. H. Gradual damping of solitary waves. Journal of Research of the National Bureau of Standards, 40 (6), 487- 498 (1948) |
| 13 | TANAKA, H., SUMER, B. M., and LODAHL, C. Theoretical and experiment investigation on laminar boundary layers under cnoidal wave motion. Coastal Engineering Journal, 40, 81- 98 (1998) |
| 14 | SUN, Y. B., ZHANG, Q. H., and ZHANG, J. F. Simulation of oscillatory laminar boundary layer flow based on lattice Boltzmann method. Journal of Hydrodynamics, 21 (3), 347- 353 (2006) |
| 15 | LIU, P. L. F., YONG, S. P., and ENWIN, A. C. Boundary layer flow and bed shear stress under a solitary wave. Journal of Fluid Mechanics, 574, 449- 463 (2007) |
| 16 | XIE, Y. Q., ZHOU, J. F., WANG, X., and DUAN, J. L. Theoretical analysis of the laminar boundary layer beneath forward-leaning waves. Coastal Engineering, 165, 103852 (2021) |
| 17 | GRANT, W. D., and MADSEN, O. S. Combined wave and current interaction with a rough bottom. Journal of Geophysical Research, 84, 1797- 1808 (1979) |
| 18 | FOSTER, D. L., GUENTHER, R. A., and HOLMAN, R. A. An analytic solution to the wave bottom boundary layer governing equation under arbitrary wave forcing. Ocean Engineering, 26, 595- 623 (1999) |
| 19 | MADSEN, O. S. Spectral wave-current bottom boundary layer flows. Proceedings of the 24th International Conference on Coastal Engineering, Coastal Engineering Research Council, Kobe, Japan, 384–398 (1994) |
| 20 | HOLMEDAL, L. E., MYRHAUG, D., and RUE, H. The sea bed boundary layer under random waves plus current. Continental Shelf Research, 23 (7), 717- 750 (2003) |
| 21 | TANAKA, H., and SHUTO, N. Friction coefficient for a wave-current coexistent system. Coastal Engineering, 24, 105- 128 (1981) |
| 22 | LARSON, M. A closed-form solution for turbulent wave boundary layers. Coastal Engineering, 1, 3244- 3256 (1996) |
| 23 | TINH, N. X., and TANAKA, H. Study on boundary layer development and bottom shear stress beneath a tsunami. Coastal Engineering Journal, 61, 574- 589 (2019) |
| 24 | NIELSEN, P. Shear stress and sediment transport calculations for swash zone modeling. Coastal Engineering, 45, 53- 60 (2002) |
| 25 | SUNTOYO, , TANAKA, H., and SANA, A. Characteristics of turbulent boundary layers over a rough bed under saw-tooth waves and its application to sediment transport. Coastal Engineering, 55, 1102- 1112 (2008) |
| 26 | TERRILE, E., RENIERS, A. J. H. M., and STIVE, M. J. F. Acceleration and skewness effects on the instantaneous bed-shear stresses in shoaling waves. Journal of Waterway, Port, Coastal, and Ocean Engineering, 135 (5), 228- 234 (2009) |
| 27 | LIN, P., CHEN, D., TANG, X., and DING, Y. Bed shear stress and turbulence characteristics under unsteady flows. Journal of Hydro-Environment Research, 21, 1- 20 (2018) |
| 28 | NIELSEN, P. 1DV structure of turbulent wave boundary layers. Coastal Engineering, 112, 1- 8 (2016) |
| 29 | TENG, Y., LU, L., CHENG, L., TONG, F., and TANG, G. A modified defect function for wave boundary layers. Coastal Engineering, 171, 104050 (2022) |
| 30 | DUNBAR, D., VAN DER, A. D. A., O'DONOGHUE, T., and SCANDURA, P. An empirical model for velocity in rough turbulent oscillatory boundary layers. Coastal Engineering, 179, 104242 (2023) |
| 31 | YUAN, J., and MADSEN, O. S. Experimental and theoretical study of wave-current turbulent boundary layers. Journal of Fluid Mechanics, 765, 480- 523 (2015) |
| 32 | ABREU, T., SILVA, P. A., SANCHO, F., and TEMPERVILLE, A. Analytical approximate wave form for asymmetric waves. Coastal Engineering, 57, 656- 667 (2010) |
| 33 | WATANABE, A. and SATO, S. A sheet-flow transport rate formula for asymmetric, forward-leaning waves and currents. Proceedings of the 29th International Conference on Coastal Engineering, National Civil Engineering Laboratory, Lisbon, 1703–1714 (2004) |
| 34 | TANAKA, H. An explicit expression of friction coefficient for wave-current coexistent motion. Coastal Engineering in Japan, 35 (1), 83- 91 (1992) |
| 35 | VAN DER, A. D. A., O'DONOGHUE, T., DAVIES, A. G., and RIBBERINK, J. S. Experimental study of turbulent boundary layer in acceleration-skewed oscillatory flow. Journal of Fluid Mechanics, 684, 251- 283 (2011) |
| 36 | YUAN, J., and MADSEN, O. S. Experimental study of turbulent oscillatory boundary layers in an oscillating water tunnel. Coastal Engineering, 89, 63- 84 (2014) |
| 37 | O'DONOGHUE, T., DAVIES, A. G., BHAWANIN, M., and VAN DER, A. D. A. Measurement and prediction of bottom boundary layer hydrodynamics under modulated oscillatory flows. Coastal Engineering, 169, 103954 (2021) |
| 38 | SUNTOYO. Study on Turbulent Bottom Boundary Layer under Nonlinear Waves and Its Application to Sediment Transport, Ph. D. dissertation, Tohoku University (2006) |
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