On the accuracy of the stationary forced Korteweg-de Vries equation as a model equation for flows over a bump
Author:
Samuel Shan Pu Shen
Journal:
Quart. Appl. Math. 53 (1995), 701-719
MSC:
Primary 76B15; Secondary 35Q53, 76B25
DOI:
https://doi.org/10.1090/qam/1359506
MathSciNet review:
MR1359506
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: Considered are the stationary supercritical free-surface solitary waves and subcritical free-surface hydraulic falls in flows over a bump in a two-dimensional channel. The stationary forced Korteweg-de Vries equation (sfKdV) sometimes can be used to predict these solitary waves and hydraulic falls. This paper studies the deviation of the results deduced from the sfKdV from the exact solutions found by computations and experiments. The elucidation of this deviation (called the error) is by comparing the sfKdV results with the computational and experimental results since an analytic approach seems impossible. From the comparisons made in this paper and the previous studies in [6], we conclude that the sfKdV is a good model in the sense that the relative error is less than 10
R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Review 18, 412–459 (1976)
A. Patoine and T. Warn, The interaction of long, quasi-stationary baroclinic waves with topography, J. Atmos. Sci. 39, 1018–1025 (1982)
T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech. 141, 455–466 (1984)
T. Y. Wu, Generation of upstream-advancing solitons by moving disturbances, J. Fluid Mech. 184, 75–99 (1987)
S. S. P. Shen, Disturbed critical surface waves in a channel of arbitrary cross section, J. Appl. Math. Phys. (ZAMP) 40, 216–229 (1989)
S. S. P. Shen, Forced solitary waves and hydraulic falls in two-layer flows, J. Fluid Mech. 234, 583–612 (1992)
S. J. Lee, G. T. Yates, and T. Y. Wu, Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid Mech. 199, 569–593 (1989)
A. S. Peters and J. J. Stoker, Solitary waves in liquids having non-constant density, Comm. Pure Appl. Math. 13, 115–164 (1960)
I. Stakgold, Green’s Functions and Boundary Value Problems, John Wiley and Sons, Chapter 9, New York, 1979
J. W. Miles, Stationary, transcritical channel flow, J. Fluid Mech. 162, 489–499 (1986)
N. S. Sivakumaran, T. Tinganchali, and R. J. Hoskings, Steady shallow flow over curved beds, J. Fluid Mech. 128, 469–487 (1983)
P. M. Naghdi and L. Vongsarnpigoon, The downstream flow beyond an obstacle, J. Fluid Mech. 162, 223–236 (1986)
F. Dias and J.-M. Vanden-Broeck, Open channel flows with submerged obstructions, J. Fluid Mech. 206, 155–170 (1989)
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2 . Comparison with experiments, J. Fluid Mech. 65, 289–314 (1974)
R. M. Miura, The Korteweg-de Vries equation: A survey of results, SIAM Review 18, 412–459 (1976)
A. Patoine and T. Warn, The interaction of long, quasi-stationary baroclinic waves with topography, J. Atmos. Sci. 39, 1018–1025 (1982)
T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech. 141, 455–466 (1984)
T. Y. Wu, Generation of upstream-advancing solitons by moving disturbances, J. Fluid Mech. 184, 75–99 (1987)
S. S. P. Shen, Disturbed critical surface waves in a channel of arbitrary cross section, J. Appl. Math. Phys. (ZAMP) 40, 216–229 (1989)
S. S. P. Shen, Forced solitary waves and hydraulic falls in two-layer flows, J. Fluid Mech. 234, 583–612 (1992)
S. J. Lee, G. T. Yates, and T. Y. Wu, Experiments and analyses of upstream-advancing solitary waves generated by moving disturbances, J. Fluid Mech. 199, 569–593 (1989)
A. S. Peters and J. J. Stoker, Solitary waves in liquids having non-constant density, Comm. Pure Appl. Math. 13, 115–164 (1960)
I. Stakgold, Green’s Functions and Boundary Value Problems, John Wiley and Sons, Chapter 9, New York, 1979
J. W. Miles, Stationary, transcritical channel flow, J. Fluid Mech. 162, 489–499 (1986)
N. S. Sivakumaran, T. Tinganchali, and R. J. Hoskings, Steady shallow flow over curved beds, J. Fluid Mech. 128, 469–487 (1983)
P. M. Naghdi and L. Vongsarnpigoon, The downstream flow beyond an obstacle, J. Fluid Mech. 162, 223–236 (1986)
F. Dias and J.-M. Vanden-Broeck, Open channel flows with submerged obstructions, J. Fluid Mech. 206, 155–170 (1989)
J. L. Hammack and H. Segur, The Korteweg-de Vries equation and water waves. Part 2 . Comparison with experiments, J. Fluid Mech. 65, 289–314 (1974)
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
76B15,
35Q53,
76B25
Retrieve articles in all journals
with MSC:
76B15,
35Q53,
76B25
Additional Information
Article copyright:
© Copyright 1995
American Mathematical Society
