Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-17T17:14:09.699Z Has data issue: false hasContentIssue false
  • English
  • Français

Mass transport under partially reflected waves in a rectangular channel

Published online by Cambridge University Press:  26 April 2006

Jiangang Wen  and
Philip L.-F. Liu
Jiangang Wen
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
Philip L.-F. Liu
Affiliation:
School of Civil and Environmental Engineering, Cornell University, Ithaca, NY 14853, USA
Get access Rights & Permissions [Opens in a new window]

Abstract

Mass transport under partially reflected waves in a rectangular channel is studied. The effects of sidewalls on the mass transport velocity pattern are the focus of this paper. The mass transport velocity is governed by a nonlinear transport equation for the second-order mean vorticity and the continuity equation of the Eulerian mean velocity. The wave slope, ka, and the Stokes boundary-layer thickness, k (ν/σ)½, are assumed to be of the same order of magnitude. Therefore convection and diffusion are equally important. For the three-dimensional problem, the generation of second-order vorticity due to stretching and rotation of a vorticity line is also included. With appropriate boundary conditions derived from the Stokes boundary layers adjacent to the free surface, the sidewalls and the bottom, the boundary value problem is solved by a vorticity-vector potential formulation; the mass transport is, in gneral, represented by the sum of the gradient of a scalar potential and the curl of a vector potential. In the present case, however, the scalar potential is trivial and is set equal to zero. Because the physical problem is periodic in the streamwise direction (the direction of wave propagation), a Fourier spectral method is used to solve for the vorticity, the scalar potential and the vector potential. Numerical solutions are obtained for different reflection coefficients, wave slopes, and channel cross-sectional geometry.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 266 , 10 May 1994 , pp. 121 - 145
Copyright
© 1994 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Bijker, E. W., Kalwijk, J. P. Th. & Pieters, T. 1974 Mass transport in gravity waves on a slopping bottom. Proc. Fourteenth Coastal Engrg. Conf. pp. 447–465. ASCE.
Dore, B. D. 1976 Double boundary layer in standing waves. Pure Appl. Geophys. 114, 629–637.Google Scholar
Haddon, E. W. & Riley, N. 1983 A note on the mean circulation in standing waves. Wave Motions 5, 43–48.Google Scholar
Hara, T. & Mei, C. C. 1987 Bragg scattering of surface waves by periodic bars: theory and experiment. J. Fluid Mech. 178, 221–241.Google Scholar
Hirasaki, G. J. & Hellums, J. D. 1968 A general formulation of the boundary conditions on the vector potential in three-dimensional hydrodynamics. Q. Appl. Maths 26, 331–342.Google Scholar
Hirasaki, G. J. & Hellums, J. D. 1970 Boundary conditions on the vector and scalar potentials in viscous three-dimensional hydrodynamics. Q. Appl. Maths 28, 293–296.Google Scholar
Iskandarani, M. & Liu, P. L.-F. 1991 Mass transport in two-dimensional water waves. J. Fluid Mech. 231, 395–415.Google Scholar
Iskandarani, M. & Liu, P. L.-F. 1993 Mass transport in a wave tank. J. Waterway, Port, Coastal, Ocean Engng 119, 88–104.Google Scholar
Liu, P. L.-F. 1977 Mass transport in the free surface boundary layer. Coastal Engng 1, 207–219.Google Scholar
Longuet-Higgins, M. S. 1953 Mass transport in water waves. Phil. Trans. R. Soc. Lond. A 245, 535–581.Google Scholar
Mei, C. C. 1989 The Applied Dynamics of Ocean Surface Waves. Advanced Series on Ocean Engineering. Vol. 1. World Scientific.
Mei, C. C. & Liu, P. L.-F. 1973 The damping of surface gravity waves in a bounded liquid. J. Fluid Mech. 59, 239–256.Google Scholar
Mei, C. C., Liu, P. L.-F. & Carter, T. G. 1972 Mass transport in water waves. Ralph M. Parson Laboratory for Water Resources and Hydrodynamics. Rep. 146. MIT, Cambridge, Massachusetts.
Riley, N. 1965 Oscillating viscous flows. Mathematika 12, 161–175.Google Scholar
Russell, R. C. H. & Osorio, J. C. D. 1958 An experimental investigation of drift profiles in a closed channel. Proc. VI Conf. on Coastal Engng. pp. 171–193. ASCE.