Laboratory experiments on waveform inversion of an internal solitary wave over a slope-shelf

Internal solitary waves (ISWs) have been detected in many parts of the world oceans, particularly over slope-shelf topography, on which signature of waveform inversion has been identified. The effects of these waves on engineering operations and ecological process have also been reported in the literature. This article reports the results of a series of numerical modeling and laboratory experiments on waveform evolution of a depression ISW in a nearly stratified two-layer fluid system, in which specific water depth ratios above the horizontal plateau of the trapezoidal obstacle were arranged to facilitate the occurrence of waveform inversion. Classifications of waveform instability (no instability, shear instability and overturning with breaking) on the slope are confirmed in the present laboratory study. Numerical results for waveform variation are also found in fair agreement with the laboratory measurements for cases without waveform inversion and minor internal breaking. Moreover, laboratory results revealed that the depth ratio of the stratified two-layer fluid above the plateau and the magnitude of the incident ISW were the two most important factors for promoting waveform inversion beyond a turning point, in addition to the requirement of a sufficient distance from the shoulder of the trapezoidal obstacle. These factors also influenced the outcome of the shoaling process, energy dissipation, internal wave breaking and turbulent mixing on the front slope, as well as the likelihood of waveform inversion on the horizontal plateau. Contrary to the common perception, it was also observed, at least from the results of the present laboratory experiments, that not all the incident ISWs of depression would produce waveform inversion on the plateau, where the upper layer was physical greater than the bottom layer, unless moderate incident wave was provided. The outcome might also be attributed to the limited distance from the shoulder onto the plateau in the present laboratory setup. However, once waveform inversion occurred on the plateau, it was found, among others, that: (1) the amplitude of the transmitted leading crest and trough might be as low as 30 and 20%, respectively, to the amplitude of the incident wave in depression; (2) the characteristic wavelength of the transmitted leading trough doubled while that of the crest was asymptotically one-half of the incident wavelength, despite the wide range variation in the depth ratios above the plateau; and (3) the transmitted potential wave energy of the leading crest contained 30% of the incident energy. Based on the results of present laboratory experiments, the range for the non-dimensional parameter α, which indicates the effect of nonlinearity and the promotion of waveform inversion on horizontal plateau, will be proposed.     

Blockage of saline intrusions in restricted,two-layer exchange flows across a submerged sill obstruction

Results are presented from a series of large-scale experiments investigating the internal and near-bed dynamics of bi-directional stratified flows with a net-barotropic component across a submerged, trapezoidal, sill obstruction. High-resolution velocity and density profiles are obtained in the vicinity of the obstruction to observe internal-flow dynamics under a range of parametric forcing conditions (i.e. variable saline and fresh water volume fluxes; density differences; sill obstruction submergence depths). Detailed synoptic velocity fields are measured across the sill crest using 2D particle image velocimetry, while the density structure of the two-layer exchange flows is measured using micro-conductivity probes at several sill locations. These measurements are designed to aid qualitative and quantitative interpretation of the internal-flow processes associated with the lower saline intrusion layer blockage conditions, and indicate that the primary mechanism for this blockage is mass exchange from the saline intrusion layer due to significant interfacial mixing and entrainment under dominant, net-barotropic, flow conditions in the upper freshwater layer. This interfacial mixing is quantified by considering both the isopycnal separation of vertically-sorted density profiles across the sill, as well as calculation of corresponding Thorpe overturning length scales. Analysis of the synoptic velocity fields and density profiles also indicates that the net exchange flow conditions remain subcritical (G < 1) across the sill for all parametric conditions tested. An analytical two-layer exchange flow model is then developed to include frictional and entrainment effects, both of which are needed to account for turbulent stresses and saline entrainment into the upper freshwater layer. The experimental results are used to validate two key model parameters: (1) the internal-flow head loss associated with boundary friction and interfacial shear; and (2) the mass exchange from the lower saline layer into the upper fresh layer due to entrainment.     

Experiments on mixing and dissipation in internal solitary waves over two triangular obstacles

A series of laboratory experiments was undertaken in a stratified two-layer fluid to investigate the energetics of the interaction between an internal solitary wave (ISW) and triangular obstacles, as well as to determine the partitioning of ISW energy and its subsequent dynamics. The ISW energy was dissipated as a result of internal breaking and turbulent mixing induced by wave instability. Tests involving different combinations of triangular obstacles in various heights and intervals and ISW of different amplitudes were performed. The wave features resulting from the interaction of an ISW and double obstacles were found to differ from those of single obstacle. The incident energy of an ISW was either reflecting back from the obstacles, dissipated through turbulent mixing, or transmitted over the double obstacles. Reduction in wave energy increased as the intervals between obstacles reduced. For two obstacles in different heights, energy dissipation was greater in the case with a higher obstacle ahead of a lower one. However, the overall performance was dependent on the relative height of the obstacles, relative water depth of the upper and bottom layer, in addition to the intervals between the obstacles.     

Using discriminant analysis to determine the breaking criterion for an ISW propagating over a ridge

This study aims to develop methods that are capable of deciding the breaking criterion for an internal solitary wave (ISW) propagating over a submarine ridge. Laboratory experiments were conducted in a wave tank to measure ISWs propagating over a submarine ridge. The results suggest that the ISW-ridge interaction can be grouped according to three degrees of magnitude based on the blockage parameter ζ and the degree of blocking B. For classification reasons, we first present an alternative decision model for evaluating the interaction of ISWs with an underwater ridge in a two-layer system. This approach is based on a multivariate statistical method and discriminant analysis. Information obtained from the eigenvalues is used to combine different ratio measures which are defined according to every single input and output. The discriminant model effectively classifies units into distinct predefined groups. An experimental simulation is conducted to demonstrate the practical implementation of the ISW-ridge interaction. The results of the method applied in this example are statistically significant, demonstrating the effectiveness of the ISW-ridge interaction classification method.     

Undular tidal bores: effect of channel constriction and bridge piers

A tidal bore may occur in a macro-tidal estuary when the tidal range exceeds 4.5–6 m and the estuary bathymetry amplifies the tidal wave. Its upstream propagation induces a strong mixing of the estuarine waters. The propagation of undular tidal bores was investigated herein to study the effect of bridge piers on the bore propagation and characteristics. Both the tidal bore profile and the turbulence generated by the bore were recorded. The free-surface undulation profiles exhibited a quasi-periodic shape, and the potential energy of the undulations represented up to 30% of the potential energy of the tidal bore. The presence of the channel constriction had a major impact on the free-surface properties. The undular tidal bore lost nearly one third of its potential energy per surface area as it propagated through the channel constriction. The detailed instantaneous velocity measurements showed a marked effect of the tidal bore passage suggesting the upstream advection of energetic events and vorticity “clouds” behind the bore front in both channel configurations: prismatic and with constriction. The turbulence patches were linked to some secondary motions and the proposed mechanisms were consistent with some field observations in the Daly River tidal bore. The findings emphasise the strong mixing induced by the tidal bore processes, and the impact of bridge structures on the phenomenon.     

Wave hydrodynamics around a multi-functional artificial reef at Leirosa

This paper describes an application of the Boussinesq-type COULWAVE model to study the wave hydrodynamics in the vicinity of a multi-functional artificial reef (MFAR). This reef is under investigation and consists of a supplementary protection solution for the Leirosa sand dune system located at South of Figueira da Foz, on the Portuguese West coast. Such installation near the coastline is expected to contribute to enhance the surfing conditions in the area, protect the sand dune system in the surroundings of Leirosa beach, and increase its environmental value. Numerical calculations with the COULWAVE model were performed for four test cases, considering two reef geometries (differing in the reef angle) and two incident wave conditions (storm condition and a common wave condition). Comparisons between the results obtained, in terms of wave heights and breaking line positions allow us to assess the influence of the reef on the hydrodynamics near the beach and around the reef. Moreover, the reef performance was analysed in terms of surfability and coastal protection. The surfability parameters (breaker height, Iribarren number and peel angle) were calculated for each test case using the numerical wave heights, wave directions and wave breaking positions. Comparisons of parameters allow characterizing the most appropriate configuration of the reef to improve the surfing conditions in the study area. A methodology based on numerical free surface elevations and horizontal velocity components was developed to calculate wave directions, since this is not a direct output of the COULWAVE model. Concerning coastal protection, analyses of the mean currents around the reef were used together with observations of the velocity cells near the shoreline as an indication of the sediment transport.     

Wave mixing rise inferred from Lyapunov exponents

We study the horizontal surface mixing and the transport induced by waves in a coastal environment. A comparative study is addressed by computing the Lagrangian coherent structures, via Finite Size Lyapunov Exponents, that arise in two different numerical settings: with and without wave coupled to currents. In general, we observe that mixing is increased in the area due to waves. Besides, the methodology presented here is tested by deploying a set of eight Lagrangian drifters at different locations. This dynamical approach is shown as a valuable tool to extract information about transport, mixing and residence embedded in the Eulerian time dependent velocity fields obtained from numerical models.     

Modelling the propagation of an internal solitary wave across double ridges and a shelf-slope

Evolution of the internal solitary waves (ISWs) in the northern South China Sea (SCS) has recently attracted the attention of many oceanographers in Taiwan and the United States. These ISWs are believed to have been induced by a branch of the Kuroshio current over Luzon Strait, which propagates westward over two ridges in the Luzon Strait between Taiwan and the Philippines, and further onto the continental margin with a shelf-slope in the SCS. This paper presents some preliminary results for the evolution of a depression ISW across two triangular obstacles using numerical modelling and laboratory experiments. The experimental results confirm that the intervals and relative height between the two obstacles are important factors in the interaction of an ISW with the obstacles. However, in the case of the movement of an ISW of depression-type across the Luzon Strait, the effect of the two ridges on the characteristics of the ISW might be less significant than that from the shelf-slope, due to the variations in relative water depth. Results from numerical experiments also show that the amplitude of an ISW can be augmented once the wave commences its contact with a shelf-slope, where an internal hydraulic jump and wave breaking with vortex motion are evident in the laboratory experiments. Eventually, an ISW of depression-type could become an elevation-type at the edge of the continental shelf landwards beyond the turning point, where the upper layer is larger than the bottom layer in a stratified water column.     

Analyzing wave breaking in a barred beach using wavelet

In this paper, a cross-shore profile evolution model, Uniform Beach Sediment Transport-Time-Averaged Cross-Shore (UNIBEST-TC), is used. The model was developed at WL/Delft hydraulic laboratory in the Netherlands. The model is used to predict wave height in a barred beach (Egmond site, The Netherlands) and the results show that there is a good agreement between the measured and predicted values by the model. In the present study, Morlet wavelet is used to distinguish the breaking waves; it is integrated over frequency to provide the temporal variation of localized total energy. The study shows that the local peaks of the energy densities correspond to the events of wave breaking in the predicted–wave time series. Furthermore, the wave energy distribution shows a tendency to decrease in the off-shore direction of the inner bar.     

Experimental study of a positive surge. Part 1: basic flow patterns and wave attenuation

A positive surge results from a sudden change in flow that increases the depth. It is the unsteady flow analogy of the stationary hydraulic jump and a geophysical application is the tidal bore. Positive surges are commonly studied using the method of characteristics and the Saint-Venant equations. The article presents the results from new experimental investigations conducted in a large rectangular channel. Detailed unsteady velocity measurements were performed with a high temporal resolution using acoustic Doppler velocimetry and non-intrusive free-surface measurement devices. Several experiments were conducted with the same initial discharge (Q = 0.060 m³/s) and six different gate openings after closure resulting in both non-breaking undular and breaking bores. The analysis of undular surges revealed wave amplitude attenuation with increasing distance of surge propagation were in agreement with Ippen and Kulin theory. Also, undular wave period and wave length data were relatively close to the values predicted by the wave dispersion theory for gravity waves in intermediate water depths.     

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

Fundamentals of nonlinear wave-particle interactions are studied experimentally in a Hele-Shaw configuration with wave breaking and a dynamic bed. To design this configuration, we determine, mathematically, the gap width which allows inertial flows to survive the viscous damping due to the side walls. Damped wave sloshing experiments compared with simulations confirm that width-averaged potential-flow models with linear momentum damping are adequately capturing the large scale nonlinear wave motion. Subsequently, we show that the four types of wave breaking observed at real-world beaches also emerge on Hele-Shaw laboratory beaches, albeit in idealized forms. Finally, an experimental parameter study is undertaken to quantify the formation of quasi-steady beach morphologies due to nonlinear, breaking waves: berm or dune, beach and bar formation are all classified. Our research reveals that the Hele-Shaw beach configuration allows a wealth of experimental and modelling extensions, including benchmarking of forecast models used in the coastal engineering practice, especially for shingle beaches.     

Dissipation of Tidal Energy and Associated Mixing in a Wide Fjord

This paper sets out to test the hypothesis that vertical mixing due to the dissipation of the internal tide accounts for a significant proportion of the total vertical mixing in a fjordic basin during a period of deep water isolation. During July and August 1999 two locations in the Clyde Sea were instrumented with moored RD Instruments Acoustic Doppler Current Profilers (ADCPs) and conductivity-temperature-pressure chains: Station C2, near the shallow entrance sill (55 m water depth), and station C1 in the deep basin (155 m water depth). A bottom pressure recorder was also deployed at station C3 by the seaward entrance to the Clyde Sea in the North Channel of the Irish Sea. A Free-falling Light Yo-yo shear microstructure profiler (FLY) was used to measure the dissipation rate of turbulent kinetic energy (TKE) throughout the water column over 25 h at both C1 and C2. These were interspersed with two-hourly conductivity-temperature-depth casts at both sites. The observations show agreement between the dissipation rate of TKE estimated by using a microstructure profiler and that estimated from the decay of the internal tide as measured by the two ADCPs. However, to account for all the implied mixing it is necessary to invoke an additional source of buoyancy flux, the most probable candidate mechanism is enhanced internal wave breaking near the sill and at the sloping boundaries of the deep basin. In addition, the vertical eddy diffusivity estimated from the micro-structure profiler (O(0.5 cm² s^–1) indicates that internal tide induced mixing away from any boundaries contributed significantly to the overall level of mixing which is required to account for the observed evolution of the deep basin water properties.     

Impulsive waves caused by subaerial landslides

This paper presents the experimental results of impulsive waves caused by subaerial landslides. A wide range of effective parameters are considered and studied by performing 120 laboratory tests. Considered slide masses are both rigid and deformable. The effects of bed slope angle, water depth, slide impact velocity, geometry, shape and deformation on impulse wave characteristics have been inspected. The impulse wave features such as amplitude, period and also energy conversation are studied. The effects of slide Froude number and deformation on energy conversation from slide into wave are also investigated. Based on laboratory measured data an empirical equation for impulse wave amplitude and period have been presented and successfully verified using available data of previous laboratory works.     

Spatial–temporal structure of mixing interface turbulence at two large river confluences

Converging flows at stream confluences often produce highly turbulent conditions. The shear layer/mixing interface that develops within the confluence hydrodynamic zone (CHZ) is characterized by complex patterns of three-dimensional flow that vary both spatially and temporally. Previous research has examined in detail characteristics of mean flow and turbulence along mixing interfaces at small stream confluences and laboratory junctions; however few, if any, studies have examined these characteristics within mixing interfaces at large river confluences. This study investigates the structure of mean velocity profiles as well as spatial and temporal variations in velocity, backscatter intensity, and temperature within the mixing interfaces of two large river confluences. Velocity, temperature, and backscatter intensity data were obtained at stationary locations within the mixing interfaces and at several cross sections within the CHZ using acoustic Doppler current profilers. Results show that mean flow within the mixing interfaces accelerates over distance from the junction apex. Turbulent kinetic energy initially increases rapidly over distance, but the rate of increase diminishes downstream. Hilbert–Huang transform analysis of time series data at the stationary locations shows that multiple dominant modes of fluctuations exist within the original signals of velocity, backscatter intensity, and temperature. Frequencies of the largest dominant modes correspond well with predicted frequencies for shallow wake flows, suggesting that mixing-interface dynamics include wake vortex shedding—a finding consistent with spatial patterns of depth-averaged velocities at measured cross sections. Spatial patterns of temperature and backscatter intensity show that the converging flows at both confluences do not mix substantially, indicating that turbulent structures within the mixing interfaces are relatively ineffective at producing mixing of the flows in the CHZ.     

A Computational Fluid Dynamics Approach for Urban Area Transport and Dispersion Modeling

A simulation tool has been developed to model the wind fields, turbulence fields, and the dispersion of Chemical, Biological, Radiological and Nuclear (CBRN) substances in urban areas on the building to city blocks scale. A Computational Fluid Dynamics (CFD) approach has been taken that naturally accounts for critical flow and dispersion processes in urban areas, such as channeling, lofting, vertical mixing and turbulence, by solving the steady-state, Reynolds-Averaged Navier–Stokes (RANS) equations. Rapid generation of high quality cityscape volume meshes is attained by a unique voxel-based model generator that directly interfaces with common Geographic Information Systems (GIS) file formats. The flow and turbulence fields are obtained by solving the steady-state RANS equations using a collocated, pressure-based approach formulated for unstructured and polyhedral mesh elements. Turbulence modeling is based upon the Renormalization Group variant of the k–ε model (k–ε RNG). Neutrally buoyant simulations are made by prescribing velocity boundary condition profiles found by a power–law relationship, while turbulence quantities boundary conditions are defined by a prescribed mixing length in conjunction with the assumption of turbulence equilibrium. Dispersion fields are computed by solving an unsteady transport equation of a dilute gas, formulated in a Eulerian framework, using the velocity and turbulence fields found from the steady-state RANS solution. In this paper the model is explained and detailed comparisons of predicted to experimentally obtained velocity, turbulence and dispersion fields are made to neutrally stable wind tunnel and hydraulic flume experiments.     

Numerical study of wave–current–vegetation interaction in coastal waters

Research on interactions among wave, current, and vegetation has received increasing attention. An explicit depth-averaged hydrodynamic model coupled with a wave spectral model (CMS-wave) was proposed in this study in order to simulate the wave and wave-induced current in coastal waters. The hydrodynamic model was based on the finite volume method while the intercell flux was computed by employing the Harten–Lax–van Leer approximate Riemann solver to investigate the dry-to-wet interface, and the drag force of vegetation was modeled as the sink terms in the momentum equations. The CMS-wave model was used to investigate the non-breaking and the breaking random waves propagation in vegetation fields. Afterwards, an empirical wave energy dissipation term with plant effect was derived to represent the resistance induced by aquatic vegetation in the wave-action balance equation. The established model was calibrated and validated with both the experimental and field data. The results showed that the wave height decreased significantly along the wave propagation direction in the presence of vegetations. The sensitivity analysis for the plant density, the wave height, and the water depth were performed by comparing the numerical results for the wave height attenuation. In addition, wave and wave-induced current through a finite patch of vegetation in the surf zone were investigated as well. The strong radiation stress gradient could be produced due to the variation of the energy dissipation by vegetation effect in the nearshore zone, which impacted the direction and amplitude of the longshore current. The calculated results showed that the coupling model had good performance in predicting wave propagation and the current over vegetated water regions.     

Fourier analysis of the roll-up and merging of coherent structures in shallow mixing layers

The current study investigates the role of nonlinearity in the development of two-dimensional coherent structures (2DCS) in shallow mixing layers. A nonlinear numerical model based on the depth-averaged shallow water equations is used to investigate temporal shallow mixing layers, where the mapping from temporal to spatial results is made using the velocity at the center of the mixing layer. The flow is periodic in the stream-wise direction and the transmissive boundary conditions are used in the cross-stream boundaries to prevent reflections. The numerical results are examined with the aid of Fourier decomposition. Results show that the previous success in applying local linear theory to shallow mixing layers does not imply that the flow is truly linear. Linear stability theory is confirmed to be only valid within a short distance from the inflow boundary. Downstream of this linear region, nonlinearity becomes important for the roll-up and merging of 2DCS. While the energy required for the merging of 2DCS is still largely provided by the velocity shear, the merging mechanism is one where nonlinear mode interaction changes the velocity field of the subharmonic mode and the gradient of the along-stream velocity profile which, in turn, changes the magnitude of the energy production of the subharmonic mode by the velocity shear implicitly. The nonlinear mode interaction is associated with energy up-scaling and is consistent with the inverse energy cascade which is expected to occur in shallow shear flows. Current results also show that such implicit nonlinear interaction is sensitive to the phase angle difference between the most unstable mode and its subharmonic. The bed friction effect on the 2DCS is relatively small initially and grows in tandem with the size of the 2DCS. The bed friction also causes a decrease in the velocity gradient as the flow develops downstream. The transition from unstable to stable flow occurs when the bed friction balances the energy production. Beyond this point, the bed friction is more dominant and the 2DCS are progressively damped and eventually get annihilated. The energy production by the velocity shear plays an important role from the upstream end all the way to the point of transition to stable flow. The fact that linear stability theory is valid only for a short distance from the inflow boundary suggests that some elements of nonlinearity is incorporated in the mean velocity profile in experiments by the averaging process. The implicit nature of nonlinear interaction in shallow mixing layers and the sensitivity of the nonlinear interaction to phase angle difference between the most unstable mode and its subharmonic allows local linear theory to be successful in reproducing features of the instability such as the dominant mode of the 2DCS and its amplitude.     

Development of a rotorcraft dust-emission parameterization using a CFD model

Flights of rotorcraft over the desert floor can result in significant entrainment of particulate matter into the atmospheric boundary layer. Continuous or widespread operation can lead to local and regional impacts on visibility and air quality. To account for this pollutant source in air quality models, a parameterization scheme is needed that addresses the complex vertical distribution of dust ejected from the rotorcraft wake into the atmospheric surface layer. A method to parameterize the wind and turbulence fields and shear stress at the ground is proposed here utilizing computational fluid dynamics and a parameterized rotor model. Measurements taken from a full scale experiment of rotorcraft flight near the surface are compared to the simulation results in a qualitative manner. The simulation is shown to adequately predict the forward detachment length of the induced ground jet compared to the measured detachment lengths. However, the simulated ground vortex widths and vorticity deviate substantially from the measured values under a range of flight speeds. Results show that the method may be applicable for air quality modeling assuming slow airspeeds of the rotorcraft, with advance ratios of 0.005–0.02.     

Kelvin–Helmholtz Instability as a Boundary-Value Problem

The Kelvin–Helmholtz (KH) instability is traditionally viewed as an initial-value problem, wherein wave perturbations of a two-layer shear flow grow over time into billows and eventually generate vertical mixing. Yet, the instability can also be viewed as a boundary-value problem. In such a framework, there exists an upstream condition where a lighter fluid flows over a denser fluid, wave perturbations grow downstream to eventually overturn some distance away from the point of origin. As the reverse of the traditional problem, this flow is periodic in time and exhibits instability in space. A natural application is the mixing of a warmer river emptying into a colder lake or reservoir, or the salt-wedge estuary. This study of the KH instability from the perspective of a boundary-value problem is divided into two parts. Firstly, the instability theory is conducted with a real frequency and complex horizontal wavenumber, and the main result is that the critical wavelength at the instability threshold is longer in the boundary-value than in the initial-value situation. Secondly, mass, momentum and energy budgets are performed between the upstream, unmixed state on one side, and the downstream, mixed state on the other, to determine under which condition mixing is energetically possible. Cases with a rigid lid and free surface are treated separately. And, although the algebra is somewhat complicated, both end results are identical to the criterion for complete mixing in the initial-value problem.