Nonlinear Wave-current Interactions Research Articles

NUMERICAL AND EXPERIMENTAL STUDY ON WAVE-CURRENT INTERACTIONS OVER A SUBMERGED BAR

Wave-current interactions are simulated by a fully-nonlinear numerical wave tank (NWT). The model is developed based on the time-domain higher-order boundary element method (HOBEM). The mixed Eulerian-Lagrangian scheme is adopted to track the transient free surface with the fourth-order Runga-Kutta method for refreshing wave profile and velocity potential at the next time step. The meshes are regridded at each time step to avoid numerical instabilities caused by mesh motions. An image Green function is utilized to increase the speed of computation. Using the developed NWT, nonlinear wave-current interactions (1) without body; (2) with a submerged vertical bar for various wave and current conditions have been simulated. The physical experiment on evolution of waves in uniform current is carried out in a wave flume at Dalian University of Technology. The flume is 69.0m long, 3.0m wide and 1.8m deep. The experimental setup is shown in Fig.1. The flume is equipped with a hydraulically driven, irregular wave generator at one end, and a wave absorber at the other. The still-water depth (h) used in this study was 0.6m, and then a vertical submerged bar was installed along the bottom of flume. It was 0.4m height and 0.5m width. In order to record the wave transformation and propagation clearly, five wave gauges were installed. The incident wave amplitude is varied from 0.025m to 0.054m, and incident wave periods are 1s and 2s, respectively. Currents are introduced by a bidirectional centrifugal pump. Following currents flow in the same direction as wave propagation, and its velocity was set for positive, and the opposite situation represented opposing currents. An acoustic Doppler velocimeter (ADV) was used to measure the current velocity and the mean velocity U0=0.2m/s was chosen. The numerical results and experimental data are compared well with each other. The effects of currents and submerged bar on wave profile, higher-harmonic waves and reflection coefficient are further studied, respectively.

Numerical investigation of the effect of current on wave focusing

In the present study, a numerical wave tank is developed to simulate the nonlinear wave-current interactions based on High Order Spectral (HOS) method. The influences of current on wave focusing are investigated by use of numerical model. The current is assumed to be constant in space. Focused waves with different amplitudes and frequency spectra are simulated with and without current. The focused wave characteristics, such as surface elevation, the maximum crest and frequency spectrum, with different current are compared. The results show that the opposing current increases the maximum crest and the energy transform during wave focusing process, and vice versa for the following current.

Interaction of finite-amplitude waves with vertically sheared current fields

A computationally efficient numerical method is developed to investigate nonlinear interactions between steep surface gravity waves and depth-varying ocean currents. The free-surface boundary conditions are used to derive a coupled set of equations that are integrated in time for the evolution of the free-surface elevation and tangential component of the fluid velocity at the free surface. The vector form of Green's second identity is used to close the system of equations. The closure relationship is consistent with Helmholtz's decomposition of the velocity field into rotational and irrotational components. The rotational component of the flow field is given by the Biot–Savart integral, while the irrotational component is obtained from an integral of a mixed distribution of sources and vortices over the free surface. Wave-induced changes to the vorticity field are modelled using the vorticity transport equation. For weak currents, an explicit expression is derived for the wave-induced vorticity field in Fourier space that negates the need to numerically solve the vorticity transport equation. The computational efficiency of the numerical scheme is further improved by expanding the kernels of the boundary and volume integrals in the closure relationship as a power series in a wave steepness parameter and using the fast Fourier transform method to evaluate the leading-order contribution to the convolution integrals. This reduces the number of operations at each time step from O(N2) to O(NlogN) for the boundary integrals and O[(NM)2] to O(NlogN) for the volume integrals, where N is the number of horizontal grid points and M is the number of vertical layers, making the model an order of magnitude faster than traditional boundary/volume integral methods. The numerical model is used to investigate nonlinear wave–current interaction in depth-uniform current fields and the modulational instability of gravity waves in an exponentially sheared current in deep water. The numerical results demonstrate that the mean flow vorticity can significantly affect the growth rate of extreme waves in narrowband sea states.

On the interaction of deep water waves and exponential shear currents

A train of periodic deep-water waves propagating on a steady shear current with a vertical distribution of vorticity is investigated by an analytic method, namely the homotopy analysis method (HAM). The magnitude of the vorticity varies exponentially with the magnitude of the stream function, while remaining constant on a particular streamline. The so-called Dubreil–Jacotin transformation is used to transfer the original exponentially nonlinear boundary-value problem in an unknown domain into an algebraically nonlinear boundary-value problem in a known domain. Convergent series solutions are obtained not only for small amplitude water waves on a weak current but also for large amplitude waves on a strong current. The nonlinear wave-current interaction is studied in detail. It is found that an aiding shear current tends to enlarge the wave phase speed, sharpen the wave crest, but shorten the maximum wave height, while an opposing shear current has the opposite effect. Besides, the amplitude of waves and fluid velocity decay over the depth more quickly on an aiding shear current but more slowly on an opposing shear current than that of waves on still water. Furthermore, it is found that Stokes criteria of wave breaking is still valid for waves on a shear current: a train of propagating waves on a shear current breaks as the fiuid velocity at crest equals the wave phase speed. Especially, it is found that the highest waves on an opposing shear current are even higher and steeper than that of waves on still water. Mathematically, this analytic method is rather general in principle and can be employed to solve many types of nonlinear partial differential equations with variable coefficients in science, finance and engineering.

Current Effects on Nonlinear Wave-Body Interactions by a 2D Fully Nonlinear Numerical Wave Tank

Nonlinear wave-current interactions with fixed or freely floating bodies are investigated by a two-dimensional (2D) fully-nonlinear numerical wave tank (NWT). The NWT is developed based on the potential theory and boundary element method (BEM) with constant panels. Mixed Eulerian-Lagrangian (MEL) time marching scheme (material-node approach) is used with fourth-order Runge-Kutta fully updated time integration, regriding, and smoothing techniques, and acceleration-potential formulation and direct mode-decomposition method. Specially devised ϕn −η type artificial damping zones (i.e., numerical beach) are implemented to prevent wave reflection from the end wall and wave maker. Using the developed NWT, nonlinear wave-current interactions (1) without bodies; (2) with a stationary body; and (3) with a floating body for various wave and current conditions have been investigated and some of the NWT simulations are compared with the results of Boussinesq’s equation and perturbation theory. It is seen that the NWT ...

Wave-induced longitudinal vortices in a shallow current

An instability theory is presented for the initiation of longitudinal vortices due to waves interacting with a turbulent current in an open channel of finite depth. With a simple model of eddy viscosity the dimensionless shearing rate of the basic current is much weaker than the dimensionless velocity almost everywhere except near the bed. We shall show that in addition to the internal vortex force, a mean shear stress exists on the mean sea level, owing to nonlinear wave–current interactions, despite the absence of wind. This stress adds vertical vorticity to the interior and augments the vortex force mechanism of instability significantly. Effects of current strength and wave conditions on the unstable growth are studied by numerical examples.

Fully nonlinear wave-current interactions and kinematics by a BEM-based numerical wave tank

A numerical wave tank (NWT) with fully nonlinear free-surface boundary conditions is developed to investigate nonlinear wave–wave and wave–current interactions and the resulting kinematics. In the present paper, the variation of wave amplitude and wave length of a monochromatic wave under several different speeds of steady uniform currents is studied through direct numerical simulations in the time domain. The nonlinear wave–current interactions are solved using a boundary integral equation method (BIEM) and a Mixed Eulerian–Lagrangian (MEL) time marching scheme. Both a semi-Lagrangian approach and Lagrangian (material-node) approach are employed and their performance is compared. A regridding algorithm based on cubic spline fitting is devised for updating the free-surface moving boundary in a stable and accurate manner. The incident waves are generated by feeding prescribed analytical waves on the input boundary. An efficient artificial numerical beach is devised and applied to dissipate wave energy and minimize wave reflections from the downstream wall. Nonlinear wave kinematics as a result of nonlinear wave–current interactions is calculated and the results are compared with a multi-layer Boussinesq model. The spatial variation of nonlinear wave profiles and kinematics affected by currents are also addressed and discussed.

Wave–current interaction with a vertical square cylinder

A three-dimensional numerical model is developed in this study to investigate the problem of wave–current–body interaction. The model solves the spatially averaged Navier–Stokes equations. Turbulence effects are modeled by a subgrid-scale (SGS) model using the concept of large eddy simulation (LES). The model is employed to study the wave–current interaction with a square cylinder that is mounted on the bottom and vertically pierces the free surface. The force analysis demonstrates that the presence of waves can reduce both the strength and frequency of vortex shedding induced by a uniform current due to the nonlinear wave–current interaction. The free surface elevation, strain rates of the mean flow, and eddy viscosity are found to closely correlate with the mechanism of vortex shedding. It is also shown that when the vortex shedding is neglected in the calculation such as by the potential flow approach, one may significantly underestimate the magnitude of in-line force. The energy spectral analysis reveals that there exist initiating, growing, and decaying regions for shedding vortices around the cylinder. In the vortex initiating region, both coherent and turbulent structures are nearly two-dimensional that become three-dimensional in the vortex growing region. The kinetic energy of both coherent and turbulent motions is dissipated in the vortex decaying region, within which the mean flow gradually returns back to two-dimensional.

A simple analytical model for surface water waves on a depth-varying current

A perturbation analysis is presented in which a series of small amplitude regular waves co-exist with an arbitrarily sheared current, U(z). Assuming that the current velocity is weak, i.e. U(z)/c=O(ϵ), the solution is extended to O(ϵ2), where c is the phase velocity and ϵ=ak the wave steepness. This provides a first approximation to the non-linear wave-current interaction, and allows simple explicit solutions for both the modified dispersion relation and the water-particle kinematics to be derived. These solutions differ from the existing irrotational models commonly used in design and, in particular, highlight the importance of the near-surface vorticity distribution. These results are shown to be in good agreement with laboratory data provided by Swan et. al. [J. Fluid Mech (2001, in press)]. Perhaps more surprisingly, good agreement is also achieved in a number of strongly non-linear wave-current combinations, where the results of the present analytical solution are compared with a fully non-linear numerical wave-current model.

Effect of nonlinear wave-current interaction on flow fields and hydrodynamic forces

A fifth-order theory for solving the problem of interaction between Stokes waves and exponential profile currents is proposed. The calculated flow fields are compared with measurements. Then the errors caused by the linear superposition method and approximate theory are discussed. It is found that the total wavccurrent field consists of pure wave, pure current and interaction components. The shear current not only directly changes the flow field, but also indirectly does so by changing the wave parameters due to wave-current interaction. The present theory can predict the wave kinematics on shear currents satisfactorily. The linear superposition method may give rise to more than 40% loading error in extreme conditions. When the apparent wave period is used and the Wheeler stretching method is adopted to extrapolate the current, application of the approximate theory is the best.

Sediment Resuspension Mechanisms in Old Tampa Bay, Florida

The mechanisms that resuspend bottom sediments in Old Tampa Bay, a shallow, microtidal, subtropical estuary in west-central Florida, were determined by analysing data collected during several periods from 1988 to 1990. Hydrodynamic and suspended-solids concentration data were collected at a relatively deep (4m) site where a permanent platform was built and at a relatively shallow (1·5m) site where a submersible instrument package was deployed. Bottom sediments were non-cohesive silts and fine sands. The primary sediment resuspension mechanism at both sites was wind waves, which were generated by strong and sustained winds associated with winter storms and tropical storms. At the platform, waves were depth-transitional, and estimated bottom shear stresses were most sensitive to wave period and water depth. Concentrations of suspended solids at this site corresponded well with wave motion, and non-linear wave-current interaction was small. At the shallow-water site, concentrations of suspended solids were elevated during periods of strong north-easterly winds and large bottom orbital velocities. At both sites, wind direction was an important factor in determining the occurrence and magnitude of sediment resuspension. Resuspended sediments settled within several hours as storm intensity diminished. Winds and waves generated by thunderstorms were more transient than those generated by winter storms and tropical storms. Based on the data collected during this study, thunderstorms are less likely to resuspend bottom sediment than winter storms and tropical storms. Maximum tidal currents at the study sites are usually less than 15cm s−1and did not increase observed concentrations of suspended solids.

Spectral Estimates of Bed Shear Stress at Subcritical Reynolds Numbers in a Tidal Boundary Layer

Velocity measurements from five levels in the constant-stress region of a neutrally stratified turbulent boundary layer are used to show that below a critical height the spectral energy density in the inertial subrange remains constant at a magnitude appropriate to the critical height. Huntley used this proposition to devise a modification to the inertial-dissipation method of estimating time-averaged bed shear stress, applicable when the velocity is measured below the critical elevation. By applying the modified method here, spectral estimates of bed stress from below the critical height were caused to collapse onto a single true value. Without the modification, the estimates of bed stress were a strong function of elevation at which the velocity was measured, which is incompatible with a constant-stress layer governed by a single velocity scale. The data are from a tidal boundary layer above an embedded wave boundary layer. The mean flow in this region experienced a hydraulic roughness that was partly attributable to nonlinear wave–current interaction in the wave boundary layer, but the realization of the turbulent-velocity spectra, from which bed shear stress was inferred, was not compromised by the wave-orbital velocities. The behavior of the spectra below the critical height is due to scale-separation effects on the energy cascade through the inertial subrange of the turbulent-velocity spectra.