Direct solution to problems of static sharp waves in shallow-water
Keywords:
the static sharp wave, shallow-water, Taylor series, Euler's equation, differential equations, potential speed, wave heightAbstract
The study of hydrodynamic canals is the first step in canals design, sediment transport, erosion, dissemination of pollution and other phenomena related to canals. When in canals, depth or flow rate is suddenly changed, the sharp wave is generated. When the position and characteristics of the sharp wave remain constant, after the steady flow, it is called the static-sharp wave. So generated jump hydraulic and sharp waves in transitions due to cross-obstacle in flow path at supercritical flows are classified as the static sharp waves. Since the equations governing the dynamics of sharp waves are the same as the flow equation in shallow-water, this paper uses linear solve for the equations governing shallow-water and specifying boundary conditions in shallow-water with the static wave and by using Taylor series, an equation for the bottom and two equations based on kinematic and dynamic boundary for the surface boundary were extracted. Also considering the frequency of the surface waves, frequency boundary conditions considering introduction of the dimensionless parameter q based on wave number and angular frequency, were extracted and other boundary conditions were rewritten based on the dimensionless parameter q. Next, based on the velocity potential and the Laplace equation, generated the differential equations solved by Euler equation, which leads to generate potential velocity (f) and wave height (h) is the canal length.
References
Athanassoulis, G.A., Makrakis, G.N., 1994. A function-theoretic approach to a two-dimensional transient wave-body interaction problem. Appl. Anal., 54, 283-303.
Berkhoff, J.C.W., 1972. Computation of combined refraction-diffraction. Proc. 13th Conf. Coastal Engng. Vancouver., vol. 2, pp. 471-490. ASCE.
Berkhoff, J.C.W., 1976. Mathematical models for simple harmonic linear waves. Wave diffraction and refraction. PhD thesis. Techn. Univ. Delft.
Berkhoff, J.C.W., Booij, N., Radder, A.C., 1982. Verification of numerical wave propagation models for simple harmonic linear water waves. Coastal Engng., 6, 255-279.
Booij, N., 1981. Gravity waves on water with non-uniform depth and current. PhD thesis, Technical University of Delft.
Booij, N., 1983. A note on the accuracy of the mild-slope equation. Coastal Engng., 7, 191-203.
Craig, W., Sulem, G., 1993. Numerical simulation of gravity waves. J. Comput. Phys., 108, 73-83.
Davies, A.G., Heathershaw, A.D., 1984. Surface-wave propagation over sinusoidal varying topography. J. Fluid Mech., 291, 419-433.
Devillard, P., Dunlop, F., Souillard, B., 1988. Localization of gravity waves on a channel with a random bottom. J. Fluid Mech., 186, 521-538.
Dingemans, M., 1997. Water Wave Propagation over Uneven Bottoms. World Scientific.
Eckart, G., 1952. The propagation of gravity waves from deep to shallow water. In Gravity Waves: Proc. NBS Semicentennial Symp. On Gravity Waves, 15-18 June 1951, pp. 156-175. Washington: National Bureau of Standards.
Euvrard, D., Jami, M., Lenoir, M., Martin, D., 1981. Recent progress towards an optimal coupling of finite elements and singularity distribution procedures in numerical ship hydrodynamics. Proc. 3rd Intl Conf. on Numer. Ship Hydrodyn. Paris. June 1981, pp. 193-210.
Evans, D.V., Kuznetsov, C.M., 1997. Gravity Waves in Water of Finite Depth (ed. J. N. Hunt), Ch. 4. Computat. Mechan. Publ.
Evans, D.V., Linton, C.M., 1994. On step approximations for water-wave problems. J. Fluid Mech., 278, 229-249.
Fitz-Gerald, G.F., 1976. The reflexion of plane gravity waves travelling in water of variable depth. Phil. Trans. R. Soc. Lond. Math. Phys. Sci., 284, 49-89.
Fitz-Gerald, G.F., Grimshaw, R.H.J., 1979. A note on the uniqueness of small-amplitude water waves travelling in a region of varying depth. Math. Proc. Camb. Phil. Soc., 86, 511-519.
Garipov, R.M., 1965. Nonsteady waves above an underwater ridge. Sov. Phys. Fluid Mech., 10, 194-196.
Hara, T. & Mei, G. C. 1987. Bragg scattering by periodic bars: theory and experiment. J. Fluid Mech. 178, 221-241.
Kirby, J.T., 1986a. A general wave equation for waves over rippled beds. J. Fluid Mech., 162, 171-186.
Kirby, J.T., 1986b. On the gradual reflection of weakly nonlinear Stokes waves in regions with varying topography. J. Fluid Mech., 162, 187-209.
Kuznetsov, N.G., 1991. Uniqueness of a solution of a linear problem for stationary oscillations of a fluid. Diff. Eqns., 27, 187-194.
Kuznetsov, N.G., 1993. The Maz'ya identity and lower estimates of eigenfrequencies of steady-state oscillations of a liquid in a channel. Russian Math. Surveys., 48, 222.
Lozano, C., Meyer, R.E., 1976. Leakage and response of wave-trapped by round islands. Phys.Fluids., 19, 1075-1088.
Massel, S., 1989. Hydrodynamics of Coastal Zones. Elsevier.
Massel, S., 1993. Extended refraction-diffraction equations for surface waves. Coastal Engng., 19, 97-126.
Mei, C.C., 1978. Numerical methods in water-wave diffraction and radiation. Ann. Rev. Fluid Mech., 10, 393-416.
Mei, C.C., 1983. The Applied Dynamics of Ocean Surface Waves. John Wiley (2nd reprint, 1994, World Scientific).
Mei, C.C., 1985. Resonant reflection of surface waves by periodic sandbars. J. Fluid Mech., 152, 315-335.
Milder, M.D., 1977. A note on Hamilton's principle for surface waves. J. Fluid Mech., 83, 159-161.
Miles, J.W., 1967. Surface-wave scattering matrix for a shelf. J. Fluid Mech., 28, 755-767.
Miles, J.W., 1991. Variational approximations for gravity waves of variable depth. J. Fluid Mech., 232, 681-688.
Moiseev, N.N., 1964. Introduction to the theory of oscillations of liquid containing bodies. Adv. Appl. Maths., 3, 233-289.
Moiseev, N.N., Rumiantsev, V.V., 1968. Dynamic Stability of Bodies Containing Fluid. Springer.
Porter, D., Chamberlain, P.G., 1997. Linear wave scattering by two-dimensional topography. Gravity Waves in Water of Finite Depth (ed. J. N. Hunt), Ch. 2. Computat. Mechan. Publ.
Radder, A.G., Dingemans, M.W., 1985. Canonical equations for almost periodic, weakly nonlinear gravity waves. Wave Motion., 7, 473-485.
Rey, V., 1992. Propagation and local behaviour of normal incident gravity waves over varying topography. Eur. J. Mech., B: Fluids 11, 213-232.
Simon, M.J., Ursell, F., 1984. Uniqueness in linearized two-dimensional water-wave problems. J. Fluid Mech., 148, 137-154.
Smith, R., Sprinks, T., 1975. Scattering of surface waves by conical island. J. Fluid Mech., 72, 373-384.
Vainberg, B.R., Maz'ja, V.G., 1973. On the problem of steady state oscillations of a fluid layer of variable depth. Trans. Moscow Math. Soc., 28, 56-73.
Yeung, R.W., 1982. Numerical methods in free-surface flows. Ann. Rev. Fluid Mech., 14, 395-442.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2014 R. Mansouri, F. Omidi Nasab, A. H. Haghiabi, B. Morshedzadeh
This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.
