Abstract
The liquid flow and the free surface shape during the initial stage of dam breaking are investigated. The method of matched asymptotic expansions is used to derive the leading-order uniform solution of the classical dam-break problem. The asymptotic analysis is performed with respect to a small parameter which characterizes the short duration of the stage under consideration. The second-order outer solution is obtained in the main flow region. This solution is not valid in a small vicinity of the intersection point between the initially vertical free surface and the horizontal rigid bottom. The dimension of this vicinity is estimated with the help of a local analysis of the outer solution close to the intersection point. Stretched local coordinates are used in this vicinity to resolve the flow singularity and to derive the leading-order inner solution, which describes the formation of the jet flow along the bottom. It is shown that the inner solution is self-similar and the corresponding boundary-value problem can be reduced to the well-known Cauchy–Poisson problem for water waves generated by a given pressure distribution along the free surface. An analysis of the inner solution reveals the complex shape of the jet head, which would be difficult to simulate numerically. The asymptotic solution obtained is expected to be helpful in the analysis of developed gravity-driven flows.
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References
Pohle FV (1950) The Lagrangian equations of hydrodynamics: solutions which are analytic functions of time. PhD dissertation, New York University, USA
Stoker JJ (1957) Water waves. Interscience Publishers Inc, New York
Korobkin AA, Pukhnachov VV (1988) Initial stage of water impact. Ann Rev Fluid Mech 20: 159–185
King AC, Needham DJ (1994) The initial development of a jet caused by fluid, body and free surface interaction. Part 1. A uniformly accelerating plate. J Fluid Mech 268: 89–101
Stansby PK, Chegini A, Barnes TCD (1998) The initial stages of dam-break flow. J Fluid Mech 374: 407–424
Zoppou C, Roberts S (2003) Explicit schemes for dam-break simulations. J Hydraul Eng 129: 11–34
Glaister P (1991) Solutions of a two dimensional dam break problem. Int J Eng Sci 29: 1357–1362
Glaister P (1988) Approximate Riemann solutions of the shallow water equations. J Hydraul Res 26: 293–300
Luigi F, Toro EF (1995) Experimental and numerical assessment of the shallow water model for two dimensional dam-break problems. J Hydraul Res 33: 843–864
Zoppou C, Roberts S (2000) Numerical solution of the two-dimensional unsteady dam break. Appl Math Fluids Res 24: 457–475
Brufau P, Garcia-Navarro P (2000) Two dimensional dam break flow simulation. Int J Numer Methods Fluids Res 33: 55–57
Hunt B (1982) Asymptotic solution for dam break problem. J Hydraul Div ASCE 108: 115–126
Penney WG, Thornhill CK (1952) The dispersion, under gravity, of a column of fluid supported on a rigid horizontal plane. Phil Trans Roy Soc London A 244: 285–311
Needham DJ, Billingham J, King AC (2007) The initial development of a jet caused by fluid, body and free surface interaction. Part 2. An impulsively moved plate. J Fluid Mech 578: 67–84
Korobkin AA (2002) Gravity driven flows. Presentation at the BAMC (unpublished)
Needham DJ, Chamberlain PG, Billingham J (2007) The initial development of a jet caused by fluid, body and free surface interaction. Part 3. An inclined accelerating plate. Q J Mech Appl Math (to appear)
Gakhov FD (1966) Boundary value problems. Pergamon Press, Oxford
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Korobkin, A., Yilmaz, O. The initial stage of dam-break flow. J Eng Math 63, 293–308 (2009). https://doi.org/10.1007/s10665-008-9237-z
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DOI: https://doi.org/10.1007/s10665-008-9237-z