Vorticity–Stream Function Formulation for Turbulent Oscillatory Boundary Layer over the Sea Bottom
Publication: Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 148, Issue 1
Abstract
The oscillatory flow within the oscillatory boundary layer at the bottom of propagating surface waves is determined by numerically solving the vorticity–stream function problem when the flow regime is turbulent. Two alternative vorticity equations are considered. The first equation is “exact” and is obtained by applying the curl operator to the Reynolds equation. The second equation is obtained by assuming that a turbulent eddy diffusivity is linearly related to the turbulent eddy viscosity and can be used to model the vorticity dynamics close to the bottom. Even though the second equation requires less computational resources and has been employed previously, it provides inaccurate results, and the use of the full equation is recommended.
Get full access to this article
View all available purchase options and get full access to this article.
Acknowledgments
This study has been partially supported by Ministero dell’Istruzione dell’Universitá e della Ricerca—MIUR (Grant No. PRIN 20172B7MY). The three authors have developed the mathematical analysis and numerical programming of the problem and written the manuscript.
References
Ambethkar, V., and M. Kumar. 2017. “Numerical solutions of 2-D steady incompressible flow in a driven square cavity using stream function-vorticity formulation.” Turk. J. Math. Comput. Sci. 6: 10–22.
Blondeaux, P., and G. Vittori. 1990. “Oscillatory flow and sediment motion over a rippled bed.” In Vol. 1 of Proc., 22nd Int. Conf. on Coastal Engineering, 2186–2199. Reston, VA: ASCE. https://doi.org/10.1061/9780872627765.167.
Blondeaux, P., and G. Vittori. 1991. “Vorticity dynamics in an oscillatory flow over a rippled bed.” J. Fluid Mech. 226: 257–289. https://doi.org/10.1017/S0022112091002380.
Blondeaux, P., G. Vittori, and G. Porcile. 2018. “Modeling the turbulent boundary layer at the bottom of sea wave.” Coastal Eng. 141 (4): 12–23. https://doi.org/10.1016/j.coastaleng.2018.08.012.
Boussinesq, J. 1987. “Essai sur la théorie des eaux courantes.” Mém. Prés. Div. Sav. Acad. Sci. 23 (1): 1–680.
Chien, J. 1979. Vorticity-stream function formulation of compressible and incompressible turbulent internal flows. Rep. No. AEDC-TR-79-13. Tullahoma, TN: Arnold Engineering Development Center, Air Force Systems Command Arnold Air Force Station.
Díaz-Carrasco, P., G. Vittori, P. Blondeaux, and M. Ortega-Sánchez. 2019. “Non-cohesive and cohesive sediment transport due to tidal currents and sea waves: A case study.” Cont. Shelf Res. 183 (8): 87–102. https://doi.org/10.1016/j.csr.2019.06.008.
Elkaim, D., M. Reggio, and R. Camarero. 1992. “Simulating two-dimensional turbulent flow by using the k–ɛ model and the vorticity-streamfunction formulation.” Int. J. Numer. Methods Fluids 14 (8): 961–980. https://doi.org/10.1002/(ISSN)1097-0363.
Jensen, B., B. Sumer, and J. Fredsoe. 1989. “Turbulent oscillatory boundary layers at high Reynolds numbers.” J. Fluid Mech. 206: 265–297. https://doi.org/10.1017/S0022112089002302.
Jones, W., and B. Launder. 1972. “The prediction of laminarization with a two-equation model of turbulence.” Int. J. Heat Mass Transfer 15 (2): 301–314. https://doi.org/10.1016/0017-9310(72)90076-2.
Kotovshchikova, M., and S. Lui. 2011. “Numerical simulation of two–dimensional flow past a dimpled cylinder using a pseudospectral method.” Appl. Math. Sci. 5 (34): 1649–1667.
Lopez, O. D., S. Pedraza, and J. R. Toro. 2017. “Simulation of axisymmetric flows with swirl in vorticity-stream function variables using the lattice Boltzmann method.” In Vortex structures in fluid dynamic problems, edited by H. Perez–de–Tejada, 95.Rijeka, Croatia: InTech.
Loukopoulos, V., G. Katsiaris, and G. Karahalios. 2003. “A steady-state solver for the simulation of Taylor vortices in spherical annular flow.” Comput. Methods Appl. Mech. Eng. 192 (26–27): 2993–3003. https://doi.org/10.1016/S0045-7825(03)00314-1.
Menter, F. 1994. “Two-equation eddy–viscosity turbulence models for engineering applications.” Am. Inst. Aeronaut. Astronaut. J. 32 (8): 1598–1605. https://doi.org/10.2514/3.12149.
Menter, F., M. Kuntz, and R. Langtry. 2003. “Ten years of industrial experience with the SST turbulence model.” Turbul. Heat Mass Transfer 4: 1–8.
Pedersen, C., R. Deigaard, J. Fredsøe, and E. A. Hansen. 1995. “Simulation of sand in plunging breakers.” J. Waterway, Port, Coastal, Ocean Eng. 121 (2): 77–87. https://doi.org/10.1061/(ASCE)0733-950X(1995)121:2(77).
Poochinapan, K. 2012. “Numerical implementations for 2D lid-driven cavity flow in stream function formulation.” Int. Sch. Res. Not. 2012 (4). https://doi.org/10.5402/2012/871538.
Ray, R. K., and J. C. Kalita. 2016. “Higher-order-compact simulation of unsteady flow past a rotating cylinder at moderate Reynolds numbers.” Comput. Appl. Math. 35 (1): 219–250. https://doi.org/10.1007/s40314-014-0191-2.
Ren, W., J. Wu, C. Shu, and W. Yang. 2012. “A stream function–vorticity formulation-based immersed boundary method and its applications.” Int. J. Numer. Methods Fluids 70 (5): 627–645. https://doi.org/10.1002/fld.v70.5.
Wilcox, D. 1988. “Reassessment of the scale-determining equation for advanced turbulent models.” Am. Inst. Aeronaut. Astronaut. J. 26 (11): 1299–1310. https://doi.org/10.2514/3.10041.
Yu, P., and Z. F. Tian. 2013. “A compact streamfunction–velocity scheme on nonuniform grids for the 2D steady incompressible Navier–Stokes equations.” Comput. Math. Appl. 66 (7): 1192–1212. https://doi.org/10.1016/j.camwa.2013.07.013.
Information & Authors
Information
Published In
Journal of Waterway, Port, Coastal, and Ocean Engineering
Volume 148 • Issue 1 • January 2022
Copyright
© 2021 American Society of Civil Engineers.
History
Received: Mar 29, 2021
Accepted: Sep 17, 2021
Published online: Nov 10, 2021
Published in print: Jan 1, 2022
Discussion open until: Apr 10, 2022
Authors
Metrics & Citations
Metrics
Citations
Download citation
If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.