Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-08T03:27:24.080Z Has data issue: false hasContentIssue false
  • English
  • Français

Forced solitary waves and hydraulic falls in two-layer flows

Published online by Cambridge University Press:  26 April 2006

Samuel Shan-Pu Shen
Samuel Shan-Pu Shen
Affiliation:
Department of Mathematics, University of Saskatchewan, Saskatoon, Canada S7N OWO Current address: Applied Mathematics Institute and Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2G1.
Get access Rights & Permissions [Opens in a new window]

Abstract

Long nonlinear waves in the two-layer flow of an inviscid, incompressible fluid are considered. Both the free surface and the interface of the two fluids are unknown free boundaries. The flow is forced by an obstruction on the bottom and/or an external pressure (usually called the wind stress) on the free surface. Away from the critical depth ratio between the surface layer and the internal layer of the fluids, the weak nonlinearity is of second order. Then there exists a balance between the dispersion and the second-order nonlinearity. The first-order asymptotic approximations of both free-surface elevation and interface elevation satisfy forced Korteweg-de Vries equations (fKdV). Because of the existence of two modes, the total number of types of solutions is double that for the single-layer flow. There are two hydraulic falls and four solitary waves for positive forcing. The first hydraulic fall and the first two solitary waves correspond to the fast mode. The remaining ones correspond to the slow mode. The first hydraulic fall was numerically found by Forbes (1989) by directly integrating the Laplace equation with nonlinear boundary conditions. There are two types of forcing according to the length of the base of the obstruction. One type is called local and the length of its base has the same scale as the height of the forcing. The expression of the forcing in the fKdV is given by the Dirac delta function. Based upon our long-wave scale the horizontal laboratory lengthscale is shrunk by ε½, while the vertical laboratory lengthscale is unchanged. Hence the semicircular bumps in the papers by Vanden-Broeck (1987), and Forbes (1988, 1989) are all considered to be local in the present paper. For the locally forced cases, stationary problems have been solved analytically. We have derived analytical expressions for the upstream speeds UL and Uc, at which the hydraulic falls can occur and solitary waves respectively cease to exist. The formulae are reduced to those due to Miles (1986) in the case of a single-layer flow and very small forcing. A full comparison among the asymptotic, computational and experimental results is provided. The comparison shows that the difference is less than 10% for UL and Uc in most of the parameter range where the asymptotic method is valid, the computational scheme converges and experiments were conducted (see figure 10). The second type of forcing is called non-local and the length of the base of the obstruction has the same scale as the wavelength. The existence and behaviour of the stationary solutions of the non-locally forced fKdV are described. Surprisingly, there can be more than four solitary waves sustained on the site of some negative forcing.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 234 , January 1992 , pp. 583 - 612
Copyright
© 1992 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

Baines, P. G. 1984 A unified description of two-layer flow over topography. J. Fluid Mech. 140, 127–167.Google Scholar
Baines, P. G. 1987 Upstream blocking and airflow over mountains. Ann. Rev. Fluid Mech. 19, 75–97.Google Scholar
Benjamin, T. B. & Lighthill, M. J. 1954 On cnoidal waves and bores. Proc. R. Soc. Lond. A 224, 448–460.Google Scholar
Cole, S. L. 1985 Transient waves produced by flow past a bump. Wave Motion 7, 579–587.Google Scholar
Dias, F. & Vanden-Broeck, J.-M. 1989 Open channel flows with submerged obstructions. J. Fluid Mech. 206, 155–170.Google Scholar
Fokas, A. S. & Ablowitz, M. J. 1989 Forced nonlinear evolution equations and the inverse scattering transform. Stud. Appl. Maths 80, 253–272.Google Scholar
Forbes, L. K. 1988 Critical free-surface flow over a semi-circular obstruction. J. Engng Maths 22, 3–13.Google Scholar
Forbes, L. K. 1989 Two-layer critical flow over a semi-circular obstruction. J. Engng Maths 23, 325–342.Google Scholar
Forbes, L. K. & Schwartz, L. W. 1982 Free-surface flow over a semi-circular obstruction. J. Fluid Mech. 114, 299–314.Google Scholar
Grimshaw, R. H. J. & Smyth, N. 1986 Resonant flow of a stratified fluid over topography. J. Fluid Mech. 169, 429–464.Google Scholar
Huang, D. B., Sibul, O. J., Webster, W. C., Wehausen, J. V., Wu, D. M. & Wu, T. Y. 1982 Ships moving in the transcritical range. In Proc. Conf. on Behavior of Ships in Restricted Waters, Varna, Vol. 2, pp. 26/1–26/10. Bulgarian Ship Hydrodynamics Center.
Hunter, J. K. & Vanden-Broeck, J.-M. 1983 Accurate computations for steep solitary waves. J. Fluid Mech. 136, 63–71.Google Scholar
Kakutani, T. & Yamasaki, N. 1978 Solitary waves on a two-layer fluid. J. Phys. Soc. Japan 45, 674–679.Google Scholar
King, A. C. & Blohr, M. I. G. 1987 Free surface flow over a step. J. Fluid Mech. 182, 193–208.Google Scholar
King, A. C. & Blohr, M. I. G. 1990 Free streamline flow over curved topography. Q. Appl. Maths 48, 281–293.Google Scholar
Lee, S. J., Yates, G. T. & Wu, T. Y. 1989 Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances. J. Fluid Mech. 199, 569–593.Google Scholar
Mei, C. C. 1986 Radiation of solitons by slender bodies advancing in a shallow channel. J. Fluid Mech. 162, 53–67.Google Scholar
Melville, W. K. & Helfrich, K. B. 1987 Transcritical two-layer flow over topography. J. Fluid Mech. 178, 31–52.Google Scholar
Miles, J. W. 1986 Stationary, transcritical channel flow. J. Fluid Mech. 162, 489–499.Google Scholar
Naghdi, P. M. & Vongsarnpigoon, L. 1986 The downstream flow beyond an obstacle. J. Fluid Mech. 162, 223–236.Google Scholar
Patoine, A. & Warn, T. 1982 The interaction of long, quasi-stationary baroclinic waves with topography. J. Atmos. Sci. 39, 1018–1025.Google Scholar
Peters, A. S. & Stoker, J. J. 1960 Solitary waves in liquids having non-constant density. Commun. Pure Appl. Maths 13, 115–164.Google Scholar
Shen, S. S. P. 1989 Disturbed critical surface waves in a channel of arbitrary cross section. Z. Angew. Math. Phys. 40, 216–229.Google Scholar
Shen, S. S. P. 1991 Locally forced critical surface waves in channels of arbitrary cross sections. Z. Angew. Math. Phys. 42, 122–128.Google Scholar
Shen, S. S. P. & Shen, M. C. 1990 A new equilibrium of subcritical flow over an obstruction in a channel of arbitrary cross section. Eur. J. Mech. B 9, 59–74.Google Scholar
Sivakumaran, N. S., Tingsanchali, T. & Hosking, R. J. 1983 Steady shallow flow over curved beds. J. Fluid Mech. 128, 469–487.Google Scholar
Smyth, N. F. 1987 Modulation theory solution for resonant flow over topography. Proc. R. Soc. Lond. A 409, 79–97.Google Scholar
Vanden-broeck, J.-M. 1987 Free-surface flow over a semi-circular obstruction in a channel. Phys. Fluids 30, 2315–2317.Google Scholar
Wu, D. M. & Wu, T. Y. 1982 Three-dimensional nonlinear long waves due to moving surface pressure. In Proc. 14th Symp. Naval Hydrodyn., pp. 125–125. Washington: National Academy of Sciences.
Wu, T. Y. 1987 Generation of upstream advancing solitons by moving disturbances. J. Fluid Mech. 184, 75–90.Google Scholar
Ym, C. S. 1979 Fluid Mechanics. Ann Arbor: West River Press.