Energy Flux Conservation Research Articles

Hydrodynamic analysis of an oscillating water column array in presence of variable bathymetry

In this study, hydrodynamic analysis of an oscillating water column (OWC) array in the presence of the variable bathymetry (seafloor depth) is performed by using a semi-analytical potential-flow solver. Within the framework of linear theory, the boundary-value problem associated with wave diffraction and radiation by an OWC array with variable bathymetry is formulated and the semi-analytical hydrodynamic model is developed using the matched eigenfunction expansion method. The fluid domain above variable bathymetry is mathematically discretized into multiple subdomains using the boundary approximation method. The Haskind relation and the energy flux conservation law are used to verify the semi-analytical model. The hydrodynamic performance of the OWC array in the presence of the ripple and coral reef bathymetries are investigated in further. Pronounced oscillations are observed for the curve of reflection coefficient Cr and hydrodynamic efficiency η for cases with ripple and coral reef bathymetries. Bragg resonance is identified as the triggering of the strong oscillations in Cr and η for case of ripple bathymetry. The primary frequency of Bragg resonance and the peaked value of Cr converge as the ripple number increases (i.e., Ns > 3). Particularly, for cases with coral reef bathymetry, the wave resonance modal in open-ended basins results in a significant wave amplification at the weather side of the OWC array.

Scattering of gravity-capillary waves on a bottom step

Under the linear approximation, we study the scattering of gravity-capillary waves on a bottom step. The boundary conditions can be satisfied by considering a countable number of evanescent modes that are localized near the bottom step. Such modes have certain specific features compared to similar modes for pure gravity waves. In particular, there exists an extra mode. When a numerical solution is calculated, this extra mode allows us to match in the fluid region and impose the additional condition at the surface, which arises for capillary gravity waves (and not for pure gravity waves). The approximate formulas are suggested for the transformation coefficients. By means of numerical calculations, we find the reflection and transmission coefficients of traveling waves and compare the results with the predictions of the approximate formulas. The approximate formulas are shown to agree well with the numerical data. Both numerical and approximate results agree with the energy flux conservation. We also study the wave transformation on the rectangular underwater bump and trench, which are straightforward extensions of our step calculations. We calculate the transformation coefficients numerically and illustrate the transformation by time-dependent results, which are also shown as animations in the supplementary material.

Time-dependent flexural gravity wave scattering due to uneven bottom in the paradigm of blocking dynamics

The present study investigates the scattering of flexural gravity waves due to uneven bottom topography in the context of wave blocking. Emphasis is given to analyzing the effects of multiple propagating wave modes on the solution procedures. The mathematical model is developed for two scenarios: a bottom step and a submerged rectangular breakwater. For the bottom step case, the complete solution in terms of the velocity potential is obtained using the eigenfunction expansion method. Subsequently, the solution associated with the wave transformation by the bottom step is extended to the case of a submerged rectangular breakwater using symmetry characteristics of the velocity potential. The energy balance relation is derived in both cases using the conservation of energy flux in the presence of multiple propagating wave modes. Wave blocking occurs for four different frequencies in both the cases of the bottom step and the submerged breakwater due to variations in water depth. This makes the problem more complex as, depending on the frequency, multiple propagating wave modes can exist in either the reflected region, the transmitted region, or both. The transmitted wave amplitude associated with the lower wavenumber within the blocking frequencies exceeds unity, and this excess energy is balanced by the corresponding energy transfer rate. Additionally, removable discontinuities are observed at the blocking frequencies in the scattering coefficients, where group velocity ceases. In the context of floating ice sheets, the deflection is analyzed in the time domain for frequencies within and outside the blocking limits.

Theoretical investigation of hydrodynamic performance of multi-resonant OWC breakwater array

In this paper, we theoretically investigate a multi-resonant oscillating water column (OWC) breakwater array. Based on linear potential flow theory, an extension of semi-analytical solution for wave interaction with periodic OWCs was developed for periodic OWCs with arbitrary chamber number. Both the diffraction and radiation problem for periodic multi-chamber OWCs are solved and verified using Haskind relation and wave energy flux conservation law. Parametrical analysis was conducted to examine the hydrodynamic characteristics of the present OWC breakwater system. Results indicate that multiple resonance of OWCs in piston mode led to multiple peaks of hydrodynamic efficiency, therefore broaden the effective frequency bandwidth with significant wave power absorption and coastal protection capability. However, the resonance of OWCs in sloshing mode in along-shore direction significantly weakens the wave power extraction. The triggering of the sloshing mode resonance mainly depends on the along-shore length of OWC unit and the incident wave angle. For larger along-shore OWC unit length (2l > h, within the present calculations), the effective frequency bandwidth is significantly influenced by the incident wave angle. Notably, when the incident wave angle exceeds π/4, the effective bandwidth decreases by half approximately. However, for narrower lengths (2l < h), the hydrodynamic performance of the OWC array demonstrates reduced sensitivity to the variations of the incident wave angle.

Flexural gravity wave interaction with an articulated heterogeneous plate within the paradigm of blocking dynamics

Surface gravity waves interact with the flexural waves to generate the flexural gravity waves whose characteristics are triggered for higher values of lateral compressive stress to generate multiple propagating wave modes. This investigation examines the scattering of obliquely incident flexural gravity waves due to articulation in two semi-infinite heterogeneous floating elastic plates in finite water depth within a blocking dynamics regime. The dispersion curve demonstrates the existence of three propagating wave modes within the blocking limits. The canonical eigenfunction expansion method used for a single propagating mode is generalized to account for multiple propagating wave modes within the limits of blocking periods. The energy relation is established using the conservation of wave energy flux and Snell's law of refraction, which depends upon the angles and amplitude of the scattered waves along with the wave energy transfer rates. The amplitude of scattering coefficients (energy transfer rate) goes beyond the unit, where the corresponding energy transfer rate (scattering coefficients) diminishes for specific wave periods. Subsequently, complete wave reflection occurs for oblique waves beyond a critical angle of incidence for a fixed period and prior to a critical angle of incidence at a higher angle of incidence. Removable discontinuities occur at the blocking and saddle points, while a jump discontinuity appears due to a change in the incident wave mode in the scattering coefficients. Surface plots reveal the irregular pattern of plate deflection for the period within the blocking limits. Linear time-dependent plate displacement is simulated in two and three dimensions.

Motion of localized disturbances in scalar harmonic lattices.

We present analytical and numerical investigations of energy propagation in systems of massive particles that interact via harmonic (linear) forces. The particle motion is described by a scalar displacement, and the particles are arranged in a simple crystal lattice. For the systems under consideration we prove the conservation of the total energy flux analytically. Then, using a sample case of a square lattice, we confirm the analytical results numerically. We create disturbances of a special kind which can move with a predefined velocity with a minor change in their shape. We show that a clot of energy, associated with each disturbance, moves similarly to a free body of matter in classical mechanics. We also numerically study a simultaneous propagation of a number of energy clots as an analogy to the motion of point masses in the conventional mechanics of particles. The obtained results demonstrate that an energy flow in lattices can be described in terms of numerous separated energy bodies, making a step towards a linkage between lattice dynamics and the kinetic theory of heat transfer in solids.

Effect of Hall current on reflection phenomenon of magneto-thermoelastic waves in a non-local semiconducting solid

The couple non-local elasticity theory is employed to discuss the reflection of magneto-thermoelastic waves in semiconductors in this paper. The effect of Hall current with the help of an external magnetic field is also studied on the propagating waves. Thermal effects in the context of heat equation of fractional order are also investigated. The Helmholtz method is used to decompose the displacement vector into its components. Four coupled waves are propagating in the medium. The amplitude ratios of the waves are computed. The conservation of energy flux is verified. The obtained results are also checked graphically under various physical parameters. Some earlier published results are successfully obtained in the absence of Hall current.

Wave‐Current Coupling Effects on the Variation Modes of Pore Pressure Response in a Sandy Seabed: Physical Modeling and Explicit Approximations

AbstractPrevious flume observations on the response of pore pressure in the seabed induced by the combined wave‐current were mainly limited to relatively deep‐water waves with wavenumber times water depth approximately above 1.0. Meanwhile, existing theoretical solutions neglected the variation of wave height induced by wave‐current coupling effects, limiting the accuracy of the predictions. In this study, the combined wave‐current induced pore pressure response within a sandy seabed is physically modeled in a water flume. A wide range of wave‐current parameters is examined. The effects of wave period, water depth, and wave height with the superimposed following and opposing currents on the change of pore pressure are investigated. The present experiments identify two new particular modes of pore pressure changing under superimposed current on waves, that is, the transition mode and the opposing‐current enhancing mode, besides the well‐recognized following‐current enhancing mode. The specific modes are determined by a dimensionless parameter characterizing the value of wavenumber times water depth for waves in the absence of a current. Based on the conservation of mass, momentum, and energy flux, an explicit solution for the wave height under the wave‐current coupling effect is derived. With this updated wave height, an analytical solution for combined wave‐current induced pore pressure response is further proposed, which agrees with the measured data. Based on this solution, a general diagram for the current effect on the mudline pore pressure amplitude is proposed, which is applicable for both laboratory and field conditions. Finally, the physical mechanism in three variation modes is discussed.

BCFT and Islands in two dimensions

By combining the AdS/BCFT correspondence and the brane world holography, we expect an equivalence relation between a boundary conformal field theory (BCFT) and a gravitational system coupled to a CFT. However, it still remains unclear how the boundary condition of the BCFT is translated in the gravitational system. We examine this duality relation in a two-dimensional setup by looking at the computation of entanglement entropy and energy flux conservation. We also identify the two-dimensional gravity which is dual to the boundary dynamics of a BCFT. Moreover, we show that by considering a gravity solution with scalar fields turned on, we can reproduce one point functions correctly in the AdS/BCFT.

Scattering of long flexural gravity wave due to structural heterogeneity in the framework of wave blocking

Scattering of flexural gravity waves due to an abrupt change in structural heterogeneity is studied within the framework of blocking dynamics in time-space under shallow water approximations. As a special case, wave scattering due to a crack in the ice sheet is obtained. In the presence of compressive force, primary and secondary wave blocking occur at two different time-periods where group velocities vanish. Moreover, for each time-period within the primary and secondary blocking points, three propagating wave modes exist of which two are related with positive group velocities and the other one corresponds to negative group velocity. Subsequently, the energy relation of the scattering problem is derived in the case of multiple propagating wave modes using the conservation of energy flux. Out of the two propagating wave modes having positive group velocities, the energy transfer rate associated with the highest wavenumber is contributing significantly to the energy relation in the vicinity of the primary blocking point, whilst the other mode behaves in a similar manner near the secondary blocking point. Moreover, between the primary and secondary blocking points the incident wave mode changes to account for law of conservation of energy flux. Further, the study reveals that the energy transfer rate and the transmission coefficient are inversely proportional to each other. In case of wave scattering due to heterogeneity in ice sheet, the reflection and transmission coefficients pattern can have more than two removable discontinuities at the blocking points, whilst in the case of wave scattering due to a crack in the ice sheet, only two removable discontinuities appear at the primary and secondary blocking points. However, a jump discontinuity occurs at the point where incident wave mode conversion takes place in both the cases of wave scattering due to heterogeneity/crack in the ice sheet. Besides, irregular behaviour in plate deflection is observed due to the superposition of multiple propagating wave modes for a certain time-period within the blocking limits.

Wave power extraction and coastal protection by a periodic array of oscillating buoys embedded in a breakwater

The integration of wave energy devices and coastal structures may be an innovative and sustainable way to achieve energy production purposes with a secondary benefit of coastal protection, which can increase accessibility and reduce the costs of wave energy technology. In this paper, a 3-D theoretical model was developed to investigate the hydrodynamic efficiency and breakwater function of a periodic array of oscillating buoys embedded in a caisson breakwater. The generalized radiation problem was solved to derive generalized radiation force. The theoretical model was validated using Haskind's relation and energy flux conservation law. The influences of wave/geometrical parameters and PTO damping were revealed. In particular, hydrodynamic phenomenon of multiple orders reflected and transmitted propagating waves and their influence on wave power extraction and coastal protection was examined. Results show that a satisfactory hydrodynamic efficiency and coastal defense are realized simultaneously under oblique waves for this proposed system. A decline of hydrodynamic efficiency is found beyond a critical wavenumber, accompanied by the occurrence of the strong reflection phenomenon. The findings of this paper contribute towards the preliminary design of the hybrid breakwater-WEC system for the synergy effect between the wave energy devices and breakwaters.

A theoretical model for an integrated wave energy extraction system consisting of a heaving buoy and a perforated wall

An integrated wave energy extraction (IWEE) system consisting of a heaving buoy and a perforated wall was investigated using the linear potential flow theory. The methods for the separation of variables and eigenfunction expansion matching were adopted to determine the spatial velocity potentials. The model was verified through wave energy flux conservation and a flume experiment. Subsequently, a comparison and a parametric study was carried out to investigate the hydrodynamic performance of the IWEE system. The introduction of the perforated wall increased the maximum capture width ratio of the buoy and significantly reduced the transmitted coefficient. Compared with the IWEE system with a solid rear wall, the proposed system mainly generated a smaller reflection coefficient and the buoy endured a smaller horizontal wave force as well. The frequency where the radiation damping equals zero was not found for the buoy of the proposed system. The wave capture efficiency and wave attenuation performance could be enhanced by properly setting the geometrical parameters. Furthermore, the wave energy captured and dissipated by the proposed IWEE system was much greater than that dissipated by an isolated perforated wall, which indicates that the system is feasible in terms of energy consumption.

Variations in Wave Slope and Momentum Flux From Wave‐Current Interactions in the Tropical Trade Winds

AbstractObservations from six Lagrangian Surface Wave Instrument Float with Tracking drifters in January‐February 2020 in the northwestern tropical Atlantic during the Atlantic Tradewind Ocean‐atmosphere Mesoscale Interaction Campaign are used to evaluate the influence of wave‐current interactions on wave slope and momentum flux. At observed wind speeds of 4–12 ms−1, wave mean square slopes are positively correlated with wind speed. Wave‐relative surface currents varied significantly, from opposing the wave direction at 0.24 ms−1 to following the waves at 0.47 ms−1. Wave slopes are 5%–10% higher when surface currents oppose the waves compared to when currents strongly follow the waves, consistent with a conservation of wave energy flux across gradients in currents. Assuming an equilibrium frequency range in the wave spectrum, wave slope is proportional to wind friction velocity and momentum flux. The observed variation in wave slope equates to a 10%–20% variation in momentum flux over the range of observed wind speeds (4–12 ms−1), with larger variations at higher winds. At wind speeds over 8 ms−1, momentum flux varies by at least 6% more than the variation expected from current‐relative winds alone, and suggests that wave‐current interactions can generate significant spatial and temporal variability in momentum fluxes in this region of prevailing trade winds. Results and data from this study motivate the continued development of fully coupled atmosphere‐ocean‐wave models.

Theoretical study on arranging heaving wave energy converters in front of jarlan-type breakwater

A hybrid wave energy extraction (HWEE) system consisting of a heaving wave energy converter (WEC) and a Jarlan-type breakwater was theoretically investigated using the linear potential theory. The law of wave energy flux conservation was used for verification. The effect of several critical geometry parameters on the hydrodynamic performance of the HWEE system was discussed. Then, appropriate parameter ranges were recommended in terms of the stability of the structure and the wave energy capture performance. The contribution of the wave radiation force to the total force was discussed, it was found that, compared with the wave excitation force, the wave radiation force can be ignored in most frequency ranges, except near the natural frequency of the heaving WEC. By a contrastive study, it was found that the hydrodynamic performance of the heaving WEC is greatly adjusted by the rear perforated breakwater, and the wave attenuation performance of the proposed HWEE system is much better than an isolated perforated breakwater. Furthermore, it was also found that, in irregular waves, the proposed HWEE system can still maintain a rather good wave energy capture performance.

A fully implicit coupled pore-network/free-flow model for the pore-scale simulation of drying processes

We present a fully coupled pore-network/free-flow model providing pore-scale insight into drying processes. We solve the Navier–Stokes equations with component transport in the free-flow region, coupled to a dynamic two-phase, two-component pore-network model (PNM) in the porous domain. The dynamic multi-physics model allows to temporally resolve drying processes in-between capillary equilibrium states. All simulations are non-isothermal and use pressure- and temperature-depended fluid properties. Carefully chosen coupling conditions and a monolithic solver ensure local conservation of mass, momentum, and energy fluxes, in particular at the interface between both model domains. We solve for wetting and non-wetting fluid pressure fields and consider advective gas transport in the network. Numerical examples demonstrate that the coupled model is able to cover a wide range of physical processes relevant for drying and show the mutual interaction of the two subregions. The model is implemented in the modular open-source framework DuMu such that extensions are straight-forward.

Propagation of periodic wave trains along the magnetic field in a collision-free plasma

In this work, a systematic study, examining the propagation of periodic and solitary waves along the magnetic field in a cold collision-free plasma, is presented. Employing the quasi-neutral approximation and the conservation of momentum flux and energy flux in the frame co-traveling with the wave, the exact analytical solution of the stationary solitary pulse is found analytically in terms of particle densities, parallel and transverse velocities, as well as transverse magnetic fields. Subsequently, this solution is generalized in the form of periodic waveforms represented by cnoidal-type waves. These considerations are fully analytical in the case where the total angular momentum flux L, due to the ion and electron motion together with the contribution due to the Maxwell stresses, vanishes. A graphical representation of all associated fields is also provided.

Meteorite impacts in the ocean: the danger of tsunamis on the coast of Buenos Aires Province, Argentina

Comets, meteorites, or asteroids impacting against the Earth are not unusual events. Such impacts on the ocean could produce tsunamis which can reach coastal areas. This paper aimed to analyze the tsunami wave heights on the coast of Buenos Aires Province produced by a meteorite impact in the South Atlantic Ocean. This subject is carried out using a simplified analytical model based on the energy flux conservation. The worst scenario was obtained in the case of the meteorite falling at the deepest continental slope edge, on a transect orientated normally to Mar del Plata coast (around 42° S–54° W). The hazard would quickly decrease if the meteorite impacted farther this location. It was also inferred that, if the meteorite fell on the Patagonian or Brazilian continental shelves, or in the Pacific, Indian or North Atlantic oceans the dangerousness would be drastically reduced. Finally, the possible implementation of this simple analytical model is analyzed in different regions of the World Ocean.

The effect of the coastal reflection on the performance of a floating breakwater-WEC system

In this paper, the hydrodynamic performance of a floating breakwater-Wave Energy Converter (WEC) integrated system was investigated under the framework of the linear potential flow theory, with a focus on the effect of the coastal reflection. The reflection coefficient and capture width ratio were verified using the rule of wave energy flux conservation. The water wave resonances, including the piston- and sloshing-mode, between the device and the coastal wall, were excited. By comparing to the cases without the coastal wall, the presence of the coastal wall led to the significant modifications of the hydrodynamic characteristics, efficiency and wave attenuation performance of the system. Specifically, the occurrence of the piston- and sloshing-mode resonance accompanies the increment of the efficiency of the system. The null efficiency was found when the device was located at the node of standing waves, which was formed due to the presence of the coastal wall. The effect of the coastal wall in irregular waves was found to be relatively milder when compared to that in regular waves.

Numerical Study on Regular Wave Shoaling, De-Shoaling and Decomposition of Free/Bound Waves on Gentle and Steep Foreshores

Numerical tests are performed to investigate wave transformations of nonlinear nonbreaking regular waves with normal incidence to the shore in decreasing and increasing water depth. The wave height transformation (shoaling) of nonlinear waves can, just as for linear waves, be described by conservation of the mechanical energy flux. The numerical tests show that the mechanical energy flux for nonlinear waves on sloping foreshores is well described by stream function wave theory for horizontal foreshore. Thus, this theory can be used to estimate the shoaled wave height. Furthermore, the amplitude and the celerity of the wave components of nonlinear waves on mildly sloping foreshores can also be predicted with the stream function wave theory. The tests also show that waves propagating to deeper water (de-shoaling) on a very gentle foreshore with a slope of cot(β) = 1200 can be described in the same way as shoaling waves. For de-shoaling on steeper foreshores, free waves are released leading to waves that are not of constant form and thus cannot be modelled by the proposed approach.

Weakly nonlinear ion sound waves in gravitational systems.

Ion sound waves are studied in a plasma subject to gravitational field giving rise to vertically inhomogeneous steady-state plasma conditions. Such systems are interesting by exhibiting a wave growth that is a result of energy flux conservation for pulses propagating in an inhomogeneous system. The increase of the amplitude of a pulse as it propagates along the density gradient in the direction of decreasing density gives rise to an enhanced interaction between waves and plasma particles that can be modeled by a modified Korteweg-de Vries equation. Analytical results are compared with numerical particle-in-cell simulations of the problem. Our code assumes isothermally Boltzmann distributed electrons resulting in a nonlinear Poisson equation. The ion component is treated as a collection of individual particles interacting through collective electric fields. Deviations from quasineutrality are allowed for.