Theoretical Solution of Long Wave Considering the Structure of Bottom Boundary Layer and Examinations on Wave Decay due to See Bottom Friction.

Cylinder wakes in shallow oscillatory flow: the coastal island wake problem

Journal of Fluid Mechanics ◽

10.1017/jfm.2019.441 ◽

2019 ◽

Vol 874 ◽

pp. 158-184 ◽

Cited By ~ 1

Author(s):

Paul M. Branson ◽

Marco Ghisalberti ◽

Gregory N. Ivey ◽

Emil J. Hopfinger

Keyword(s):

Boundary Layer ◽

Oscillatory Flow ◽

Bottom Friction ◽

Stability Parameter ◽

Bottom Boundary Layer ◽

Ekman Pumping ◽

Bottom Boundary ◽

Island Wakes ◽

Island Wake ◽

The Stability

Topographic complexity on continental shelves is the catalyst that transforms the barotropic tide into the secondary and residual circulations that dominate vertical and cross-shelf mixing processes. Island wakes are one such example that are observed to significantly influence the transport and distribution of biological and physical scalars. Despite the importance of island wakes, to date, no sufficient, mechanistic description of the physical processes governing their development exists for the general case of unsteady tidal forcing. Controlled laboratory experiments are necessary for the understanding of this complex flow phenomenon. Here, three-dimensional velocity field measurements of cylinder wakes in shallow-water oscillatory flow are conducted across a parameter space that is typical of tidal flow around shallow islands. The wake form in steady flows is typically described in terms of the stability parameter $S=c_{f}D/h$ (where $D$ is the island diameter, $h$ is the water depth and $c_{f}$ is the bottom boundary friction coefficient); in tidal flows, there is an additional dependence on the Keulegan–Carpenter number $KC=U_{0}T/D$ (where $U_{0}$ is the tidal velocity amplitude and $T$ is the tidal period). In this study we demonstrate that when the influence of bottom friction is confined to a Stokes boundary layer the stability parameter is given by $S=\unicode[STIX]{x1D6FF}^{+}/KC$ where $\unicode[STIX]{x1D6FF}^{+}$ is the ratio of the wavelength of the Stokes bottom boundary layer to the depth. Three classes of wake form are observed with decreasing wake stability: (i) steady bubble for $S\gtrsim 0.1$; (ii) unsteady bubble for $0.06\lesssim S\lesssim 0.1$; and (iii) vortex shedding for $S\lesssim 0.06$. Transitions in wake form and wake stability are shown to depend on the magnitude and temporal evolution of the wake return flow. Scaling laws are developed to allow upscaling of the laboratory results to island wakes. Vertical and lateral transport depend on three parameters: (i) the flow aspect ratio $h/D$; (ii) the amplitude of tidal motion relative to the island size, given by $KC$; and (iii) the relative influence of bottom friction to the flow depth, given by $\unicode[STIX]{x1D6FF}^{+}$. A model of wake upwelling based on Ekman pumping from the bottom boundary layer demonstrates that upwelling in the near-wake region of an island scales with $U_{0}(h/D)KC^{1/6}$ and is independent of the wake form. Finally, we demonstrate an intrinsic link between the dynamical eddy scales, predicted by the Ekman pumping model, and the island wake form and stability.

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NEAR BOTTOM VELOCITIES IN WAVES WITH A CURRENT; ANALYTICAL AND NUMERICAL COMPUTATIONS

Coastal Engineering Proceedings ◽

10.9753/icce.v19.79 ◽

1984 ◽

Vol 1 (19) ◽

pp. 79 ◽

Cited By ~ 2

Author(s):

W.G.M. Van Kesteren ◽

W.T. Bakker

Keyword(s):

Boundary Layer ◽

Three Dimensional ◽

Bottom Friction ◽

Bottom Boundary Layer ◽

Two Dimensional ◽

Layer Model ◽

Bottom Boundary ◽

Waves And Currents ◽

A Current ◽

Bottom Velocities

In this paper, starting from the Prandtl hypothesis a three-dimensional numerical bottom boundary layer model has been developed, which allows to calculate bottom friction by a combination of waves and currents. The model has been compared with two-dimensional analytical computations which gave similar results. The bottom friction values found are comparable to the ones, found by Lundgren (1972), however in the most relevant cases somewhat less. Furthermore in the two-dimensional case the model has been compared with measurements of Bakker and Van Doom (1978). With respect to the oscillatory motion, still some minor deviations occur between theory and measurements, due to deficiencies of the Prandtl theory.

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