The initial development of a jet caused by fluid, body and free-surface interaction. Part 1. A uniformly accelerating plate

The present study treats the three-dimensional hydrodynamic slamming problem on a vertical plate subjected to the impact of a steep wave moving towards the plate with a constant velocity. The problem is complicated significantly by assuming that there is a rectangular opening on the plate which allows the discharge of the liquid. The analysis is conducted analytically assuming linear potential theory. The examined configuration determines two boundary value problems with mixed conditions which should be taken into account properly. The mathematical process assimilates the plate with a degenerate elliptical cylinder allowing thus the employment of elliptical harmonics that ensure the satisfaction of the free-surface boundary condition of the front face of the steep wave, beyond the plate. This assumption leads to an additional boundary value problem with mixed conditions in the vertical direction. The associated problem involves triple trigonometrical series and it is solved through a transformation into integral equations. To tackle the boundary value problem in the vertical direction a perturbation technique is employed. Extensive numerical calculations are presented as regards the variation of the velocity potential on the plate at the instant of the impact which reveals the influence of the opening. The theory is extended to the computation of the total impulse exerted on the plate using pressure-impulse theory.

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“…! Equation (2.34) is explicitly identical with the 2D strip theory solution 6 { 0, 5 provided by King & Needham (1994).…”

Section: Fig 1 the Geometry Of The Impact Problemmentioning

confidence: 92%

Three-dimensional steep wave impact on a vertical cylinder

2016

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“…We are unaware of attempts to construct the inner solutions of the dam-break problem either in Lagrangian or Eulerian variables. However, such an inner solution was successfully derived in a relevant problem concerning a uniformly accelerating wavemaker by King and Needham [4]. In the present study, we use the methodology and findings from [4], but we employ a formal small parameter ε, which characterizes the small duration of the initial stage under consideration.…”

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confidence: 99%

“…In a two-dimensional formulation the latter problem was studied by King and Needham [4] for a uniformly accelerating plate within a small-time approximation. King and Needham suggested an asymptotic procedure which allowed them to describe the initial development of a jet caused by the accelerating plate.…”

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confidence: 99%

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The initial stage of dam-break flow

Korobkin

Yılmaz

2008

J Eng Math

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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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“…Assuming potential flow, in which the velocity potential φ has been made dimensionless with U h, x and z with h and t with h/U , and in the absence of vortex sheets, we have to solve where η may well be multi-valued, with The initial conditions are appropriate to the jet collision problem under discussion. Hence, over this timescale, we have an 'unsteady Helmholtz flow' for which, as discussed in King & Needham (1994) even the small time solution is very complicated. Nonetheless, we anticipate that this solution will describe the growth of a vertical column of liquid centred near the initial impact point and that, as t → ∞ with F = ∞, the flow will tend to a steady Helmholtz flow in a frame moving with the spout.…”

Section: Discussionmentioning

confidence: 99%

Non-classical shallow water flows

Edwards¹,

Howison²,

Ockendon³

et al. 2007

IMA Journal of Applied Mathematics

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This paper deals with violent discontinuities in shallow water flows with large Froude number F .On a horizontal base, the paradigm problem is that of the impact of two fluid layers in situations where the flow can be modelled as two smooth regions joined by a singularity in the flow field. Within the framework of shallow water theory we show that, over a certain timescale, this discontinuity may be described by a delta-shock, which is a weak solution of the underlying conservation laws in which the depth and mass and momentum fluxes have both delta function and step function components. We also make some conjectures about how this model evolves from the traditional model for jet impacts in which a spout is emitted.For flows on a sloping base, we show that for flow with an aspect ratio of O(F −2 ) on a base with an O(1) or larger slope, the governing equations admit a new type of discontinuous solution that is also modelled as a deltashock. The physical manifestation of this discontinuity is a small 'tube' of fluid bounding the flow. The delta-shock conditions for this flow are derived and solved for a point source on an inclined plane. This latter delta-shock framework also sheds light on the evolution of the layer impact on a horizontal base.

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The initial development of a jet caused by fluid, body and free-surface interaction. Part 1. A uniformly accelerating plate

Cited by 48 publications

References 5 publications

Three-dimensional steep wave impact on a vertical cylinder

Three-dimensional steep wave impact on a vertical cylinder

The initial stage of dam-break flow

Non-classical shallow water flows

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