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Modelling of coastal and nearshore structures and processes
Published in P. Novak, V. Guinot, A. Jeffrey, D.E. Reeve, Hydraulic Modelling – an Introduction, 2010
P. Novak, V. Guinot, A. Jeffrey, D.E. Reeve
As computer hardware has gained in power and our understanding of wave processes has increased, so the sophistication of numerical wave-transformation models has developed. A significant step in this development was the introduction of the mild-slope equation by Berkoff (1972). The mild-slope equation is derived from the linearized governing equations of irrotational flow in three dimensions under the assumption that the bottom varies slowly over the scale of a wavelength. The mild-slope equation has been used widely to date to predict wave properties in coastal regions. The equation, which can deal with generally complex wave fields with satisfactory accuracy, accounts for refraction, shoaling and diffraction (and, in some forms, reflection as well). The mild-slope equation may be written as () ∇⋅(ccg∇Φ)+ω2cgΦc=0
Design wave specification
Published in Dominic Reeve, Andrew Chadwick, Christopher Fleming, Coastal Engineering, 2018
Dominic Reeve, Andrew Chadwick, Christopher Fleming
A significant step in the development was the introduction of the mild-slope equation, first derived by Berkhoff (1972). The mild-slope equation is derived from the linearised governing equations of irrotational flow in three dimensions under the assumption that the bottom varies slowly over the scale of a wavelength. The mild-slope equation has been used widely to date to predict wave properties in coastal regions. The equation, which can deal with generally complex wave fields with satisfactory accuracy, accounts for refraction, shoaling and diffraction (and in some forms reflection as well). The mild-slope equation may be written as ∇.(ccg∇Φ)+ω2cgΦc=0 for the complex two-dimensional potential function Φ. In a three-dimensional Cartesian coordinate system, Φ is related to the water wave velocity potential of linear periodic waves φ(x,y,z,t) by Φ(x,y)=φ(x,y,z,t)cosh(κh)cosh(κ(h+z))e−iωtwhere the frequency ω is a function of the wavenumber k = (k,l) with κ = |k| by virtue of the dispersion relationship ω2=gκtanh(κh).
Barrier and bottom topography effects on hydroelastic response of floating elastic plate in a two-layer fluid
Published in Geophysical & Astrophysical Fluid Dynamics, 2023
N. M. Prasad, R. M. Prasad, Prashant Kumar, Pulkit Kumar, Chandra Mani Prasad
Among all the variety of analytical approaches for the uniform bottom topography, one that is famously known is the method of eigenfunctions expansion (MEE). However, in the case of an arbitrary bottom profile, analytical solutions are rare due to the complex Robin-type boundary condition (8) on the bottom. To overcome this difficulty, a semi-analytical approach was proposed by Chamberlain and Porter (2005) under the assumption that bottom has mild-slope i.e. , where h is water depth and λ is wave-length. The approximation has been used by solving a mild-slope equation derived from variational principle. The posed physical problem is tackled with mild-slope equation for the variable bottom topography whereas MEE for the uniform bottom topography. Solution in terms of an algebraic system of equations is derived by applying the matching and jump conditions at the interfaces. Further, the plate edge conditions are also applied to study plate response to the incident waves. It is to be noted that only propagating modes associated with free-surface and interface waves are used for the solution because for two-layer fluid model with bottom undulations, the mild-slope equation which incorporates the eigen modes related to all evanescent waves is not available in the literature. For solution, the eigenfuctions associated with free surface (z=0) and interface (z=−h) waves through the index notation n=1 and n=2 respectively, are represented below. In the following, the expansions of velocity potentials in each fluid region and the associated physical parameters are illustrated.
Surface gravity wave interaction with a partial porous breakwater in a two-layer ocean having bottom undulations
Published in Waves in Random and Complex Media, 2021
R. B. Kaligatla, S. Tabssum, T. Sahoo
Of various physical processes associated with wave transformation, there is significant growth in the understanding of refraction-diffraction in surface gravity waves due to variations in bottom topography since the pioneering work of Berkhoff [4]. The investigation was carried out by deriving suitable mild-slope equations through the depth-averaging procedure. In its model context, the mild-slope equation is a typically simplified differential equation to analyze the combined effects of refraction and diffraction of waves and offers an inexpensive technique. First, Rojanakamthorn et al. [5] derived a mild-slope equation for exploring the performance of submerged porous trapezoidal breakwater mounted on a sloping bed. As a further study of this work, Losada et al. [6] analyzed the kinematics and dynamics outside the water region and inside the porous breakwater by extending the mild-slope equation for oblique wave transmission. Shu and Park [7] and Zhu and Chwang [8] described an application of the modified mild-slope equation for a sloping plane bottom to study wave trapping by a complete porous breakwater. Further, Zhu [9] used an extended mild-slope equation for the model of porous breakwater lying on sea-bottom undulations. Silva et al. [10] presented a generic version of the mild-slope equation for plane waves propagating over a porous bed whose thickness is arbitrarily finite and demonstrated the earlier known results in individual cases. Subsequently, the effects of evanescent waves were also taken into account by Silva et al. [11]. Kaligatla et al. [12] explored the performance of a breakwater system having multiple thin porous barriers in an ocean with a step-type sloping bottom. Further, the influence of bottom undulations on wave scattering and trapping by a rectangular submerged porous breakwater can be found in the study of Tabssum et al. [13].
Study on the influence of infragravity waves on inundation characteristics at Minami-Ashiyahama in Osaka Bay induced by the 2018 Typhoon Jebi
Published in Coastal Engineering Journal, 2020
Naohiro Hattori, Yoshimitsu Tajima, Yusuke Yamanaka, Kenzou Kumagai
Secondly, the characteristics of harbor-scale resonance with period of less than 300 s was analyzed. Since the focusing wave period is shorter than the ones for the former analysis of bay-scale resonance, linear mild slope equations (Tajima et al. 2016b) were applied for this numerical study. A linear mild slope equation corresponds to the linear shallow water equation under the limit of long wave conditions. The computation domain is nearly the same as the area shown in Figure 2, and the grid size and time step was set at 10 m and 0.25 s. Incident wave angle was also set to SSW, following the disaster report by Hyogo Prefecture (2019b). Similar to the numerical study on bay-scale resonance, the ratio of local and incident wave heights excluding the shoaling effect, R, was computed and obtained R was also plotted as a function of incident wave period as shown in the panels (d-1) of Figures 10 and 11. For computation of the shoaling coefficient, the water depth at the offshore boundary is set to 22.8 m and the depth at Kobe, Osaka, and Minami-Ashiyahama is set to 16 m, 14 m, and 15.5 m respectively. At Kobe, peaks can be seen at approximately 80 s, 170 s, and 220 s, which matches well with the observed data in Figure 10 (d-2). At Osaka, a peak can be seen at approximately 125 s, which is a peak that can be seen clearly from observation data in Figure 11 (d-2). The frequencies of these peaks seen at Kobe and Osaka do not match with each other, and this feature indicates that these peak frequencies between 1/300 Hz and 1/80 Hz should be due to harbor-scale resonance dominantly affected by local layout of coastal structures. At Kobe shown in Figure 10, a relatively high R is observed at 30 s < T < 40 s in the panel (d-1) while no clear peaks were observed in (d-2). This difference may be because of the filtering effect of the tide gauge station at Kobe.