Bottom Ripple Research Articles

Mathematical modeling of a submerged piezoelectric wave energy converter device installed over an undulated seabed

A piezoelectric wave energy converter (PWEC) device integrated with an impermeable breakwater placed over an undulated seabed is considered. The PWEC device is composed of a single submerged flexible plate with piezoelectric layers attached to both faces of the flexible plate. Due to the piezoelectric effect, this piezoelectric plate generates electricity when excited by the incident waves. A detailed analysis is done to investigate the effect of PWEC plate submergence depth, plate length, plate edge conditions, incident wave period, bottom ripple amplitude, and ripple number on the power generation by the PWEC device. It is seen that the PWEC device edge conditions, submergence depth, and plate length play a significant role in the resonating pattern associated with the wave power generation curve. The results demonstrate that the PWEC device having moderate plate length and with free and moored type front edges generates a higher amount of wave power for a wider range of incident wave frequencies.

Upstream-propagating waves induced by steady current over a rippled bottom: theory and experimental observation

Abstract

Explosive instability due to flow over a rippled bottom

Abstract. In this paper, we study Bragg resonance, i.e., the triad interaction between surface and/or interfacial waves with a bottom ripple, in the presence of background velocity. We show that when one of the constituent waves of the triad has negative energy, the amplitudes of all the waves grow exponentially. This is very different from classic Bragg resonance in which one wave decays to cause the growth of the other. The instabilities we observe are “explosive” and are different from normal mode shear instabilities since our velocity profiles are linearly stable. Our work may explain the existence of large-amplitude internal waves over periodic bottom ripples in the presence of tidal flow observed in oceans and estuaries.

Irregular wave transmission on bottom bumps using fully nonlinear NURBS numerical wave tank

In this study, a fully nonlinear three-dimensional numerical wave tank (NWT) is developed to simulate propagation of nonlinear random sea wave over the bottom ripple patches. Evolution of the free surface is performed by the non-uniform rational B-spline formulation (NURBS) and the mixed Eulerian–Lagrangian method (MEL). A high-order boundary integral equation is used to solve the Laplace equation in the Eulerian frame. The free surface is updated by the material node method and the fourth-order Runge–Kutta time integration scheme. Appropriate numerical solutions are obtained by deploying damping zones at the both tank end walls. Also, the NURBS approximation is applied to compute the kinematics of the free surface particles. Propagation of the irregular waves in an NWT is conducted and compared with the available experimental and numerical data. Bragg reflection of the random sea wave due to bottom ripple patches are compared with the prior numerical studies. The fully nonlinear free surface evolution of transmitted spectrum due to presence of the hemispherical bumps is modeled successfully. Also fifth-order Stokes wave is applied as a strong nonlinear incident wave to interact with the hemispherical bottom bumps.

Observation of Anomalous Resistance Behavior in Bilayer Graphene

Our measurement results have shown that bilayer graphene exhibits an unexpected sharp transition of the resistance value in the temperature region 200~250 K. We argue that this behavior originates from the interlayer ripple scattering effect between the top and bottom ripple graphene layer. The inter-scattering can mimic the Coulomb scattering but is strongly dependent on temperature. The observed behavior is consistent with the theoretical prediction that charged impurities are the dominant scatters in bilayer graphene. The resistance increase with increasing perpendicular magnetic field strongly supports the postulate that magnetic field induces an excitonic gap in bilayer graphene. Our results reveal that the relative change of resistance induced by magnetic field in the bilayer graphene shows an anomalous thermally activated property.

Parameterization of bottom friction under combined wave-tide action in the Hooghly estuary, India

A comprehensive knowledge of the wave climate is an essential pre-requisite for practical applications in coastal and near-shore region. In coastal areas the dominant dissipative mechanism appears to be bottom friction where the relative strength depends on bottom characteristics, sediment type and bottom ripple geometry. The present state-of-art version SWAN wave model uses three popular formulations for dissipative mechanism due to bottom interaction. The bottom friction coefficient as in SWAN cannot be a constant and needs to be tuned based on prevailing hydrodynamic conditions when applied for a tidal dominant region. A combined wave-current interaction study should take into account varying water levels, reversal of current system and associated shear on the sea-bed. The present work reports on development of a new resistance law for bottom friction under the combined action of waves and currents. This new formulation has been implemented in the SWAN model and further its implication on variability of significant wave height for the Hooghly estuary located in the Bay of Bengal, East coast of India was investigated by comparison with ENVISAT satellite measurement. Based on this study, it could be ascertained that new friction formulation has potential for wind-wave modeling studies in near-shore and coastal waters.

Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples

We consider a class of higher order (quartet) Bragg resonance involving two incident wave components and a bottom ripple component (so called class III Bragg resonance). In this case, unlike class I/II Bragg resonance involving a single incident wave and one/two bottom ripple components, the frequency of the resonant wave, which can be reflected or transmitted, is a sum or difference of the incident wave frequencies. In addition to transferring energy across the spectrum leading to potentially significant spectral transformation, such resonances may generate long (infragravity) waves of special importance to coastal processes and engineering applications. Of particular interest here is the case where the incident waves are oblique to the bottom undulations (or to each other) which leads to new and unexpected wave configurations. We elucidate the general conditions for such resonances, offering a simple geometric construction for obtaining these. Perturbation analysis results are obtained for these resonances predicting the evolutions of the resonant and incident wave amplitudes. We investigate special cases using numerical simulations (applying a high-order spectral method) and compare the results to perturbation theory: infragravity wave generation by co- and counter-propagating incident waves normal to bottom undulations; longshore long waves generated by (bottom) oblique incident waves; and propagating–standing resonant waves due to (bottom) parallel incident waves. Finally, we consider a case of multiple resonance due to oblique incident waves on bottom ripples which leads to complex wave creation and transformations not easily tractable with perturbation theory. These new wave resonance mechanisms can be of potential importance on continental shelves and in littoral zones, contributing to wave spectral evolution and bottom processes such as sandbar formation.

Utilizing High-Frequency Acoustic Backscatter to Estimate Bottom Sand Ripple Parameters

In some applications of underwater acoustics, it is important to know the ripple structure on shallow-water sediments. For example, the prediction of buried target detection via sound scattering by ripples depends critically on the ripple height and spatial wavelength. Another example is the study of sediment transport, where knowing the ripple structure and its evolution over time helps to understand the forcing on the bottom and the response of sediments. Here, backscatter data from a 300-kHz system are used to show that ripple wavelength and height can be estimated from backscatter images via a simple inversion formula. The inversion results are consistent with in situ measurements of the ripple field using an independent measurement system. Motivated by the backscatter data, we have developed a time-domain numerical model to simulate scattering of high-frequency sound by a ripple field. This model treats small-scale scatterers as Lambertian scatterers distributed randomly on the large-scale ripple field. Numerical simulations are conducted to investigate the conditions under which remote sensing of bottom ripple heights, wavelength, and its power spectrum is possible.

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part I. Perturbation analysis

We investigate, via perturbation analyses, the mechanisms of nonlinear resonant interaction of surface-interfacial waves with a rippled bottom in a two-layer density-stratified fluid. As in a one-layer fluid, three classes of Bragg resonances are found to exist if nonlinear interactions up to the third order in the wave/ripple steepness are considered. As expected, the wave system associated with the resonances is more complicated than that in a one-layer fluid. Depending on the specifics of the resonance condition, the resonance-generated wave may be a surface or internal mode and may be transmitted or reflected. At the second order, class I Bragg resonance occurs involving two surface and/or internal waves and one bottom-ripple component. The interaction of an incident surface/internal wave with the bottom ripple generates a new surface or internal wave that may propagate in the same or the opposite direction as the incident wave. At the third order, class II and III Bragg resonances occur involving resonant interactions of four wave/ripple components: two surface and/or internal waves and two bottom-ripple components for class II resonance; three surface and/or internal waves and one bottom-ripple components for class III resonance. As in class I resonance, the resonance-generated wave in class II resonance has the same frequency as that of the incident wave. For class III resonance, the frequency of the resonant wave is equal to the sum or difference of the two incident wave frequencies. We enumerate and represent, using Feynman-like diagrams, the possible cases and combinations for Bragg resonance up to the third order (in two dimensions). Analytical regular perturbation results are obtained and discussed for all three classes of Bragg resonances. These are valid for limited bottom patch lengths and initial/finite growth of the resonant waves. For long bottom patches, a uniformly valid solution using multiple scales is derived for class I resonance. A number of applications underscoring the importance and implication of these nonlinear resonances on the evolution of ocean waves are presented and discussed. For example, it is shown that three internal/surface waves co-propagating over bottom topography are resonant under a broad range of Bragg conditions. The present study provides the theoretical basis and understanding for the companion paper (Alam, Liu & Yue 2008), where a direct numerical solution for the general nonlinear problem is pursued.

Bragg resonance of waves in a two-layer fluid propagating over bottom ripples. Part II. Numerical simulation

We develop a direct numerical method to study the general problem of nonlinear interactions of surface/interfacial waves with variable bottom topography in a two-layer density stratified fluid. We extend a powerful high-order spectral (HOS) method for nonlinear gravity wave dynamics in a homogeneous fluid to the case of a two-layer fluid over non-uniform bottom. The method is capable of capturing the nonlinear interactions among large number of surface/interfacial wave mode and bottom ripple components up to an arbitrary high order. The method preserves exponential convergence with respect to the number of modes of the original HOS and the (approximately) linear effort with respect to mode number and interaction order. The method is validated through systematic convergence tests and comparison to a semi-analytic solution we obtain for an exact nonlinear Stokes waves on a two-layer fluid (in uniform depth). We apply the numerical method to the three classes of generalized Bragg resonances studied in Alam, Liu & Yue (J. Fluid Mech., vol. 624, 2009, p. 225), and compare the perturbation predictions obtained there with the direct simulation results. An important finding is possibly the important effect of even higher-order nonlinear interactions not accounted for in the leading-order perturbation analyses. To illustrate the efficacy of the numerical method to the general problem, we consider a somewhat more complicated case involving two incident waves and three bottom ripple components with wavenumbers that lead to the possibility of multiple Bragg resonances. It is shown that the ensuing multiple (near) resonant interactions result in the generation of multiple new transmitted/reflected waves that fill a broad wavenumber band eventually leading to the loss of order and chaotic motion.

Subcritical grazing angle detection of targets buried under a rippled sand interface

Controlled acoustic measurements were conducted to investigate subcritical detection of targets buried in sand having a rippled sediment-water interface. The measurements were performed in a freshwater test-pond with a 1.5-m layer of sand on the bottom. Bottom ripple was formed by scraping the sand with a machined rake. Broadband (10 to 50 kHz) transducers were placed onto an elevated rail that translated horizontally permitting data to be processed using synthetic aperture sonar (SAS) techniques. Targets included silicone oil-filled spheres and flat-endcapped, solid aluminum cylinders. Data were acquired for various ripple wavelengths, heights, and orientations. For each bottom configuration, the seabed roughness over the buried target was measured to verify the ripple parameters and to estimate the small-scale roughness spectrum. This information was used in models predicting backscatter signal levels from the target and bottom. Results for spherical targets have generally been good. However, when cylindrical targets were inspected, model agreement was variable for reasons not fully understood. Characteristics of the target return are found to depend sensitively on ripple height and wavelength, and bottom parameters. Results are discussed with an eye towards understanding previously unexplained field test detections of buried cylindrical targets. [Work supported by ONR and SERDP.]

Comparison of measured buried target backscatter levels with high-order predictions: Rippled interface effects

Controlled sonar measurements with buried spheres and cylinders have demonstrated diffraction by bottom ripple can significantly improve target detection performance with sonar operated at grazing angles shallower than the critical grazing angle. Furthermore, buried target scattering models based on low-order perturbation theory to handle sound transmission through bottom roughness confirm that a significant enhancement in detection performance can be expected. But comparisons between measurements and model predictions have not always produced good agreement. As the bottom ripple height increases, discrepancies have been noted in comparisons with the predicted backscatter levels from a buried sphere, suggesting a failure of low-order perturbation theory to properly account for high-amplitude roughness effects. Recently, a recursive algorithm to generate arbitrary-order perturbation corrections for transmission and reflection through an idealized sinusoidal ripple was formulated and incorporated into our existing buried target scattering codes. The resulting higher-order predictions indicate previous discrepancies were not necessarily due to the neglect of high-order corrections. Calculations show model inputs such as bottom attenuation can influence backscatter levels significantly and measurement of these parameters may need to be performed more carefully. We demonstrate this through a presentation of data and model comparisons. [Work supported by ONR and SERDP.]

Bragg scattering of surface waves over permeable rippled beds with current

In this study we develop a time-dependent wave equation for waves propagating with a current over permeable rippled beds. As well known, Bragg resonance occurs when the incident wavelength is twice the wavelength of the bottom ripple undulation and no current is present. However, the current in the near-shore region changes the resonance condition. A one-dimensional wave field is solved numerically based on the derived equation to study the effect of current on the Bragg resonance condition. Nonlinear wave–wave resonant interaction theory provides an explanation of the effect on Bragg resonance. Numerical results also indicate that the maximum reflection coefficient increases as current velocity increases from a negative to a positive value. Furthermore, the velocity of the current affects the position of the maximum reflection coefficient.

Authors' reply

A comprehensive knowledge of the wave climate is an essential pre-requisite for practical applications in coastal and near-shore region. In coastal areas the dominant dissipative mechanism appears to be bottom friction where the relative strength depends on bottom characteristics, sediment type and bottom ripple geometry. The present state-of-art version SWAN wave model uses three popular formulations for dissipative mechanism due to bottom interaction. The bottom friction coefficient as in SWAN cannot be a constant and needs to be tuned based on prevailing hydrodynamic conditions when applied for a tidal dominant region. A combined wave-current interaction study should take into account varying water levels, reversal of current system and associated shear on the sea-bed. The present work reports on development of a new resistance law for bottom friction under the combined action of waves and currents. This new formulation has been implemented in the SWAN model and further its implication on variability of significant wave height for the Hooghly estuary located in the Bay of Bengal, East coast of India was investigated by comparison with ENVISAT satellite measurement. Based on this study, it could be ascertained that new friction formulation has potential for wind-wave modeling studies in near-shore and coastal waters.