Water Waves Over a Channel of Finite Depth With a Submerged Plane Barrier

A depth-resolved framework for the coupling of sub-surface flows and surface gravity water waves

10.5194/egusphere-egu2020-20360 ◽

2020 ◽

Author(s):

Simen Ådnøy Ellingsen ◽

Stefan Weichert ◽

Yan Li

Keyword(s):

Surface Waves ◽

Water Waves ◽

Multiple Scales ◽

Subsurface Flow ◽

Vertical Gradient ◽

Finite Depth ◽

Surface Gravity ◽

Linear Wave ◽

Flow Equations ◽

Direct Integration Method

This work aims to develop a new framework for the interaction of a subsurface flow and surface gravity water waves, based on a perturbation and multiple-scales expansion.&#160; Surface waves are assumed of a narrow band &#948; (&#948; ), indicating they can be expressed as a carrier wave whose amplitude varies slowly in space and time relative to its phase. Using the Direct Integration Method proposed in Li & Ellingsen (2019), the effects of the vertical gradient of a subsurface flow are taken into account on the linear wave properties in an implicit fashion. At the second order in wave steepness &#1013;, the forcing of the sub-harmonic bound waves is considered that plays a role in the primary equations for a subsurface flow.The novel framework derives the continuity and momentum equations for a subsurface flow in two different formats, including both the depth integrated as well as the depth resolved version. The former compares with Smith (2006) to examine the roles of the rotationality of wave motions in the subsurface flow equations. The latter employs the sigma coordinate system proposed in Mellor (2003, 2008, 2015) and extends the framework therein to allow for quasi-monochromatic surface waves and the effects of the shear of a current on linear surface waves. Compared to Mellor (2003, 2008, 2015), the vertical flux/vertical radiation stress term in the proposed framework is approximated to one order of magnitude higher, i.e. O(&#1013;2&#948;2).ReferencesLi, Y., Ellingsen, S. &#197;. A framework for modeling linear surface waves on shear currents in slowly varying waters.&#160;J. Geophys. Res. C: Oceans,&#160;(2019) 124(4), 2527-2545.Mellor, G. L. The three-dimensional current and surface wave equations. J. Phys. Oceanogr., (2003) 33, 1978&#8211;1989.Mellor, G. L. The depth-dependent current and wave interaction equations: a revision.&#160;J. Phys. Oceanogr.,&#160;(2008) 38(11), 2587-2596.Smith, J. A. Wave&#8211;current interactions in finite depth.&#160;Journal of Physical Oceanography,&#160;(2006) 36(7), 1403-1419.

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An explicit Hamiltonian formulation of surface waves in water of finite depth

Journal of Fluid Mechanics ◽

10.1017/s0022112092003483 ◽

1992 ◽

Vol 237 ◽

pp. 435-455 ◽

Cited By ~ 21

Author(s):

A. C. Radder

Keyword(s):

Surface Waves ◽

Water Waves ◽

Hamiltonian Formulation ◽

Vertical Plane ◽

Finite Depth ◽

Short Wave ◽

Linear Dispersion ◽

Energy Functionals ◽

Canonical Variables ◽

Wave Nonlinearity

A variational formulation of water waves is developed, based on the Hamiltonian theory of surface waves. An exact and unified description of the two-dimensional problem in the vertical plane is obtained in the form of a Hamiltonian functional, expressed in terms of surface quantities as canonical variables. The stability of the corresponding canonical equations can be ensured by using positive definite approximate energy functionals. While preserving full linear dispersion, the method distinguishes between short-wave nonlinearity, allowing the description of Stokes waves in deep water, and long-wave nonlinearity, applying to long waves in shallow water. Both types of nonlinearity are found necessary to describe accurately large-amplitude solitary waves.

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The scattering of surface waves by a vertical plane barrier

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410004514x ◽

1969 ◽

Vol 66 (2) ◽

pp. 417-422 ◽

Cited By ~ 11

Author(s):

R. J. Jarvis ◽

B. S. Taylor

Keyword(s):

Surface Waves ◽

Small Amplitude ◽

Vertical Plane ◽

Finite Depth ◽

Infinite Depth ◽

Plane Barrier

AbstractIn this paper we use a method due to Williams(1) to discuss the scattering of surface waves of small amplitude on water of infinite depth by a fixed vertical plane barrier extending indefinitely downwards from a finite depth.

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Forcing of a bottom-mounted circular cylinder by steep regular water waves at finite depth

Journal of Fluid Mechanics ◽

10.1017/jfm.2014.386 ◽

2014 ◽

Vol 755 ◽

pp. 1-34 ◽

Cited By ~ 82

Author(s):

Bo T. Paulsen ◽

H. Bredmose ◽

H. B. Bingham ◽

N. G. Jacobsen

Keyword(s):

Free Surface ◽

Circular Cylinder ◽

Water Waves ◽

Finite Depth ◽

Load Cycle ◽

Harmonic Force ◽

Third Harmonic ◽

The Third ◽

Nonlinear Motion ◽

Secondary Load

AbstractForcing by steep regular water waves on a vertical circular cylinder at finite depth was investigated numerically by solving the two-phase incompressible Navier–Stokes equations. Consistently with potential flow theory, boundary layer effects were neglected at the sea bed and at the cylinder surface, but the strong nonlinear motion of the free surface was included. The numerical model was verified and validated by grid convergence and by comparison to relevant experimental measurements. First-order convergence towards an analytical solution was demonstrated and an excellent agreement with the experimental data was found. Time-domain computations of the normalized inline force history on the cylinder were analysed as a function of dimensionless wave height, water depth and wavelength. Here the dependence on depth was weak, while an increase in wavelength or wave height both lead to the formation of secondary load cycles. Special attention was paid to this secondary load cycle and the flow features that cause it. By visual observation and a simplified analytical model it was shown that the secondary load cycle was caused by the strong nonlinear motion of the free surface which drives a return flow at the back of the cylinder following the passage of the wave crest. The numerical computations were further analysed in the frequency domain. For a representative example, the secondary load cycle was found to be associated with frequencies above the fifth- and sixth-harmonic force component. For the third-harmonic force, a good agreement with the perturbation theories of Faltinsen, Newman & Vinje (J. Fluid Mech., vol. 289, 1995, pp. 179–198) and Malenica & Molin (J. Fluid Mech., vol. 302, 1995, pp. 203–229) was found. It was shown that the third-harmonic forces were estimated well by a Morison force formulation in deep water but start to deviate at decreasing depth.

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An experimental study of the pressure variations in standing water waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1951.0079 ◽

1951 ◽

Vol 206 (1086) ◽

pp. 424-435 ◽

Cited By ~ 22

Keyword(s):

Experimental Study ◽

Surface Waves ◽

Standing Wave ◽

Water Waves ◽

Wave Length ◽

Second Order ◽

First Order ◽

Standing Water ◽

Plane Barrier ◽

Progressive Waves

Although the first-order pressure variations in surface waves on water are known to decrease exponentially downwards, it has recently been shown theoretically that in a standing wave there should be some second-order terms which are unattenuated with depth. The present paper describes experiments which verify the existence of pressure variations of this type in waves of period 0·45 to 0·50 sec. When the motion consists of two progressive waves of equal wave-length travelling in opposite directions, the amplitude of the unattenuated pressure variations is found to be proportional to the product of the wave amplitudes. This property is used to determine the coefficient of reflexion from a sloping plane barrier.

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DISPERSION IN SEISMIC SURFACE WAVES

Geophysics ◽

10.1190/1.1437652 ◽

1951 ◽

Vol 16 (1) ◽

pp. 63-80 ◽

Cited By ~ 26

Author(s):

Milton B. Dobrin

Keyword(s):

Surface Waves ◽

Water Waves ◽

Seismic Refraction ◽

Wave Theory ◽

Wave Dispersion ◽

Surface Zone ◽

Surface Wave Dispersion ◽

Near Surface ◽

Arrival Times ◽

Ground Roll

A non‐mathematical summary is presented of the published theories and observations on dispersion, i.e., variation of velocity with frequency, in surface waves from earthquakes and in waterborne waves from shallow‐water explosions. Two further instances are cited in which dispersion theory has been used in analyzing seismic data. In the seismic refraction survey of Bikini Atoll, information on the first 400 feet of sediments below the lagoon bottom could not be obtained from ground wave first arrival times because shot‐detector distances were too great. Dispersion in the water waves, however, gave data on speed variations in the bottom sediments which made possible inferences on the recent geological history of the atoll. Recent systematic observations on ground roll from explosions in shot holes have shown dispersion in the surface waves which is similar in many ways to that observed in Rayleigh waves from distant earthquakes. Classical wave theory attributes Rayleigh wave dispersion to the modification of the waves by a surface layer. In the case of earthquakes, this layer is the earth’s crust. In the case of waves from shot‐holes, it is the low‐speed weathered zone. A comparison of observed ground roll dispersion with theory shows qualitative agreement, but it brings out discrepancies attributable to the fact that neither the theory for liquids nor for conventional solids applies exactly to unconsolidated near‐surface rocks. Additional experimental and theoretical study of this type of surface wave dispersion may provide useful information on the properties of the surface zone and add to our knowledge of the mechanism by which ground roll is generated in seismic shooting.

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Asynchronous whirl in a rotating cylinder partially filled with liquid

Journal of Fluid Mechanics ◽

10.1017/s0022112085000143 ◽

1985 ◽

Vol 150 ◽

pp. 311-327 ◽

Cited By ~ 16

Author(s):

A. S. Berman ◽

T. S. Lundgren ◽

A. Cheng

Keyword(s):

Surface Waves ◽

Shallow Water ◽

Solitary Waves ◽

Water Waves ◽

Rotating Cylinder ◽

The Self ◽

Hydraulic Jumps ◽

Centrifugal Forces ◽

Shallow Water Waves ◽

Analytical Results

Experimental and analytical results are presented for the self-excited oscillations that occur in a partially filled centrifuge when centrifugal forces interact with shallow-water waves. Periodic and aperiodic modulations of the basic whirl phenomena are both observed and calculated. The surface waves are found to be hydraulic jumps, undular bores or solitary waves.

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