Effective medium approximation model of sea foam layer microwave emissivity of a vertical profile

A sea foam layer produced by wave breaking consists of seawater-coated air bubbles, fluid water, and air. The non-uniformity and microstructure of the air–water mixture in a foam layer can cause some important effects on microwave or optical properties. Considering the vertical non-uniformity of the air–water volume, the air-volume fraction is derived as a function of the foam depth variable, coated air bubble velocity, and foam temperature using the gas convection–diffusion equation. For a vertical graded profile of the air volume fraction, we discuss the effects of the coated air bubble velocity parameter and the air–sea temperature difference on the air volume fraction. The results show that the coated air bubble velocity is a key parameter that widely modulates the air volume fraction. Furthermore, an effective medium approximation (EMA) of spherical shell microstructures is proposed to investigate the graded foam effective permittivity and the emissivity of the sea surface covered with the foam layer of the graded air volume fraction in a vertical profile, and good agreement is obtained on comparing the EMA results with the experimental data of foam layer microwave emissivities at frequencies 1.4, 10.8, and 36.5 GHz. In our EMA model, the depth average of graded foam permittivity is adopted. This model can produce reasonable results by only tuning the air bubble velocity. It indicates that the EMA model can combine the effects of graded foam permittivity and foam microstructures into the coated air bubble velocity parameter. Meanwhile, we have qualitatively discussed the influence of air–sea temperature difference on the brightness temperature of the foam layer. It is shown that a negative (or positive) air–sea temperature difference enhances (or reduces) the sea surface brightness temperature, and the brightness temperature difference increases with an increase in microwave frequency.     

Simulation of spilling breaking waves using a two phase flow CFD model

A two phase flow CFD model has been developed for 2D spilling breaking wave simulations. A mass conservative level set method similar to Olsson and Kreiss [Olsson E, Kreiss G. A conservative level set method for two phase flow. J Comput Phys 2005;210(1):225–46] is implemented for capturing the air–water interface. The solver is discretised using a finite volume method based on a curvilinear coordinate system. A fully implicit fractional step method is used to advance simulations in time. The solver has been tested and validated by repeating benchmark results of dam breaking simulation and travelling solitary wave simulation. Finally, we employ this solver to simulate spilling breaking waves in the surf zone. Our results show that surface elevations, the location of the breaking point and undertow profiles can generally be well captured. We have also found that temporal and spatial schemes may have significant impacts on computational results.     

Simulation of ship wakes image by an along-track interferometric SAR

A mathematical model that allows simulations of the image of waves in ship wakes by either regular or interferometric airborne Synthetic Aperture Radar (SAR or INSAR) is described. The three-component velocity field induced at the ocean surface by a moving ship serves as the input to the model. The simulations take into account the effect of temporal variations of the wave field in the wake on the INSAR imaging by the velocity bunching mechanism. The model also accounts for the scanning distortion of the image. The developed algorithm allows study of the visibility of the ship wave wake by a regular SAR or by INSAR for arbitrary imaging parameters, as well as for different ship sizes and ship velocity vectors relative to the platform flight track. Various patterns of ship wake images obtained by numerical simulations are presented.     

C3: A finite volume-finite difference hybrid model for tsunami propagation and runup

A numerical model that couples Finite Difference and Finite Volume schemes has been developed for tsunami propagation and runup study. An explicit leap-frog scheme and a first order upwind scheme has been considered in the Finite Difference module, while in the Finite Volume scheme a Godunov Type method based on the f-waves approach has been used. The Riemann solver included in the model corresponds to an approximate augmented solver for the Shallow Water Equations (SWE) in the presence of variable bottom surface. With this hybrid model some of the problems inherent to the Godunov type schemes are avoided in the offshore region, while in the coastal area the use of a conservative method ensures the correct computation of the runup and wave breaking. The model has been tested and validated using different problems with a known analytical solution and also with laboratory experiments, considering both non breaking and breaking waves. The results are very satisfactory, showing that the hybrid approach is a useful technique for practical usages.     

A fixed-grid model for simulation of a moving body in free surface flows

A two-dimensional computer model is developed to simulate free surface flow interaction with a moving body. The model is based on the cut-cell technique in a fixed-grid system. In this model, a body is approximated by the partial cell treatment (PCT), in which an irregular body is represented by the volumetric fraction of solid in Cartesian cells. The body motion is tracked by Lagrangian method whereas the fluid motion around the body is solved by Eulerian method. The concept of “locally relative stationary (LRS)” is introduced in this study. In the LRS method, a source term is added locally to the conventional continuity equation on body surfaces to take account of body motions, which subsequently affects the computational results of fluid pressure and flow velocity around the body. The LRS method is incorporated into an earlier Reynolds averaged Navier-Stokes (RANS) equations model developed by Lin and Liu [A numerical study of breaking waves in the surf zone. J Fluid Mech 1998;359:239-64]. The new model is capable of simulating generic turbulent free surface flows and their interaction with a moving body or multiple moving bodies. A series of numerical experiments have been conducted to verify the accuracy of the model for simulation of moving body interaction with a free surface flow. These tests include the generation of a solitary wave with the prescribed wave paddle movements, water exit and water impact and entry of a horizontal circular cylinder, fluid sloshing in a horizontally excited tank, and the acceleration/deceleration of an elliptical cylinder near a water surface. Excellent agreements are obtained when numerical results are compared to available analytical, experimental, and other numerical results. The model is a simple-to-implement computational tool for simulating a moving body in turbulent free surface flows.     

Unsteady RANS simulation of the ship forward speed diffraction problem

The forward speed diffraction problem for a surface ship is analyzed numerically, using a RANS approach with a single-phase level set method to compute the free surface and a blended k-ε/k-ω model for the turbulent viscosity. Simulations were run for a DTMB 5512 model under head incident waves at two speeds and two wavelengths with the same wave amplitude (a = 0.006L, with L the ship length). The medium speed case (Fr = 0.28) with long wavelength incident waves (λ = 1.5L) behaves linearly and has been extensively compared against available experimental data for resistance and heave forces and pitching moment, unsteady free surface elevations, and unsteady velocity fields at the nominal wake plane (x/L = 0.935). Quantitative verification and validation was performed for this case by running three grids and three time steps with refinement ratio of and the flow field analyzed in detail. The behavior of the boundary layer is analyzed to explain the origin of large first harmonic amplitudes on the axial velocity observed both experimentally and numerically. The high speed case (Fr = 0.41) with short wavelength incident waves (λ = 0.5L) exhibits non-linear behavior on the forces and moment with a strong second harmonic component and an unsteady breaking bow wave. The second harmonic has been reproduced by the CFD computations and the breaking wave predicted. Analysis of the flow indicates that the breaking wave could be responsible for the non-linear behavior on the forces and moment.     

Discontinuous Galerkin Method for Linear Free-Surface Gravity Waves

In this paper, we discuss a discontinuous Galerkin finite (DG) element method for linear free surface gravity waves. We prove that the algorithm is unconditionally stable and does not require additional smoothing or artificial viscosity terms in the free surface boundary condition to prevent numerical instabilities on a non-uniform mesh. A detailed error analysis of the full time-dependent algorithm is given, showing that the error in the wave height and velocity potential in the L²-norm is in both cases of optimal order and proportional to O(Δt²+h^p+1), without the need for a separate velocity reconstruction, with p the polynomial order, h the mesh size and Δt the time step. The error analysis is confirmed with numerical simulations. In addition, a Fourier analysis of the fully discrete scheme is conducted which shows the dependence of the frequency error and wave dissipation on the time step and mesh size. The algebraic equations for the DG discretization are derived in a way suitable for an unstructured mesh and result in a symmetric positive definite linear system. The algorithm is demonstrated on a number of model problems, including a wave maker, for discretizations with accuracy ranging from second to fourth order.This revised version was published online in July 2005 with corrected volume and issue numbers.     

Computations of Explosive Boiling in Microgravity

Dynamics of the explosive growth of a vapor bubble in microgravity is investigated by direct numerical simulation. A front tracking/finite difference technique is used to solve for the velocity and the temperature field in both phases and to account for inertia, viscosity, and surface deformation. The method is validated by comparison of the numerical results with the available analytical formulations such as the evaporation of a one-dimensional liquid/vapor interface, frequency of oscillations of capillary waves, and other numerical results. Evolution of a three-dimensional vapor nucleus during explosive boiling is followed and a fine scale structure similar to experimental results is observed. Two-dimensional simulations yield a similar qualitative instability growth.     

A quantitative sub-grid air entrainment model for bubbly flows - plunging jets

The accurate prediction of air entrainment is critical in simulating various important multiphase (air/water) flows. In this paper, we present a sub-grid air entrainment model that quantitatively predicts the rate of air entrainment and subsequent disperse bubbly flow for a plunging jet. The derivation of this model is based on the two-stage (i.e., low and high liquid jet velocity) air entrainment mechanisms suggested by Sene [Sene KJ. Air entrainment by plunging jets. Chem Eng Sci 1988;43(10):2615-23]. This model was validated against extensive experimental data for water jets in air over a wide range of liquid velocities (from around 1 to 10 m/s) for the total rate of air entrainment. It was then implemented into an Eulerian/Eulerian two-fluid computational multiphase fluid dynamics (CMFD) model, wherein the liquid and the bubbles are modeled as two distinct continua. This multiphase model, supplemented by the new sub-grid air entrainment model, was used to predict the void fraction distribution underneath plunging water jets at different depths and water jet velocities. It was found that this approach yields results that match the experimental observations very well.     

A multiple-layer σ-coordinate model for simulation of wave-structure interaction

A 3D multiple-layer σ-coordinate model has been developed to simulate surface wave interaction with various types of structures including submerged structures, immersed structures, and floating structures. This model is the extension of the earlier model [Lin P, Li CW. A σ-coordinate three-dimensional numerical model for surface wave propagation. Int J Numer Methods Fluid 2002;38(11):1045-68] that solves Navier-Stokes equations in the transformed σ-coordinate, which is especially efficient for simulation of wave propagation over varying topography. By introducing the layered σ-coordinates, the present model overcomes the difficulty encountered by the earlier model in calculating waves past a depth discontinuity, e.g., a submerged rectangular breakwater. Furthermore, with the employment of 3-layer σ-coordinate the present model is able to simulate flow interaction with an immersed body or a floating body. The new model is validated against an established Volume-Of-Fluid (VOF) model [Lin P, Liu PL-F. A numerical study of breaking waves in the surf zone. J Fluid Mech 1998;359:239-64] for the 2D solitary wave interaction with a submerged, immersed, or floating rectangular obstacle. For the solitary wave interaction with a submerged breakwater, the numerical results are also compared to the experimental data by Zhuang and Lee [A viscous rotational model for wave overtopping over marine structure. In Proc 25th Int Conf Coast Eng, ASCE, 1996. p. 2178-91] and very good agreements have been obtained for velocities in the vortex behind the structure. Finally, the present model is used to simulate 3D wave interaction with a Very Large Floating Structure (VLFS) above a submerged shoal. It is proved that the model is an accurate and efficient numerical tool to investigate different wave-structure interactions problems.     

Numerical Simulation of growth and collapse of a bubble induced by a pulsed microheater

A three-dimensional numerical analysis of the growth and collapse of a bubble on a microheater is presented. SIMULENT code, which solves the full Navier-Stokes equations with surface tension effects, is used in these simulations. A volume of fluid (VOF) interface tracking algorithm is used to track the evolution of the free surface flow. A one-dimensional heat conduction model is used to consider the energy transfer between the bubble and the surrounding liquid, as well as the temperature distribution in the liquid layer. Details of the velocity and pressure distribution in the liquid during the growth and collapse of the vapor bubble are obtained. Numerical results for the growth and the collapse of the bubbles are compared with those of experiments under similar conditions. Comparisons show that the volume evolution of the vapor bubble is well predicted by the numerical model.     

Travelling waves in a reaction-diffusion model for electrodeposition

In this paper we consider an analytical and numerical study of a reaction-diffusion system for describing the formation of transition front waves in some electrodeposition (ECD) experiments. Towards this aim, a model accounting for the coupling between morphology and composition of one chemical species adsorbed at the surface of the growing cathode is addressed. Through a phase-space analysis we prove the existence of travelling waves, moving with specific wave speed. The numerical approximation of the PDE system is performed by the Method of Lines (MOL) based on high order space semi-discretization by means of the Extended Central Difference Formulae (D2ECDF) introduced in [1]. First of all, to show the advantage of the proposed schemes, we solve the well-known Fisher scalar equation, focusing on the accurate approximation of the wave profile and of its speed. Hence, we provide numerical simulations for the electrochemical reaction-diffusion system and we show that the results obtained are qualitatively in good agreement with experiments for the electrodeposition of Au-Cu alloys.     

Explaining extreme waves by a theory of stochastic wave groups

It is well known that in a Gaussian sea an extreme wave event is a particular realization of the space–time evolution of a well defined linear wave group, in agreement with the theory of quasi-determinism of Boccotti [Boccotti P. On mechanics of irregular gravity waves. Atti Acc Naz Lincei, Memorie 1989;19:11–170] and the Slepian model of Lindgren [Kac M, Slepian D. Large excursions of Gaussian processes. Ann Math Statist 1959;30:1215–28; Lindgren G. Some properties of a normal process near a local maximum. Ann Math Statist 1970;4(6):1870–83]. In this paper, the concept of stochastic wave groups is proposed to explain the occurrence of extreme waves in nonlinear random seas, according to the dynamics imposed by the Zakharov equation [Zakharov VE. Statistical theory of gravity and capillary waves on the surface of a finite-depth fluid. J Eur Mech B—Fluids 1999;18(3):327–44]. As a corollary, a new analytical solution for the probability of exceedance of the crest-to-trough height is derived for the prediction of extreme wave events in nonlinearly modulated long-crested narrow-band seas. Furthermore, a generalization of the Tayfun distribution [Tayfun MA. On narrow-band representation of ocean waves. Part I: Theory. J Geophys Res 1986;91(C6):7743–52] for the wave crest height is also provided. The new analytical distributions explain qualitatively well recent experimental results of Onorato et al. [Onorato M, Osborne AR, Cavaleri L, Brandini C, Stansberg CT. Observation of strongly non-Gaussian statistics for random sea surface gravity waves in wave flume experiments. Phys Rev E 2004;70:067302] and the numerical simulations of Socquet-Juglard et al. [Socquet-Juglard H, Dysthe K, Trulsen K, Krogstad HE, Liu J. Probability distributions of surface gravity waves during spectral changes. J Fluid Mech 2005;542:195–216].     

On the application of the single-phase level set method to naval hydrodynamic flows

The application of the single-phase level set approach to the numerical simulations of three-dimensional free surface flows around complex geometries, at both non-breaking and breaking regimes is presented. In this approach only the liquid phase is simulated and the level set function is used as tracking device to locate the free surface position. The extrapolation of the solution in the dummy points in the gaseous phase is such that second-order accuracy is maintained also in the points adjacent to the free surface; the time evolution of the level set function and the re-initialization step have been merged so to get a function which is a distance function everywhere, and satisfies, at the same time, the kinematic condition on the free surface. The implementation of this technique into a general purpose Reynolds averaged Navier-Stokes (RANS) equations solver developed at INSEAN [Di Mascio A, Broglia R, Favini B. A Second Order Godunov-type Scheme for Naval Hydrodynamics. Kluwer Academic/Plenum Publishers; 2001, p. 253-61], is described in details; capabilities of the algorithm in dealing with non-breaking and breaking flows in the naval hydrodynamic context will be demonstrated by using a submerged hydrofoil and two different ship hulls in straight course as test cases. Comparisons with both experimental data and numerical surface fitting computations are presented; convergence properties of the algorithm, as well as validation and verification assessment will be also discussed.     

Effects of submersion depth on wave uplift force acting on Biloxi Bay Bridge decks during Hurricane Katrina

A large portion of the Biloxi Bay Bridge was submerged and destroyed by surface waves and storm surge associated with Hurricane Katrina in 2005. In this paper, the time history of wave forces exerted on the Biloxi Bay Bridge during Hurricane Katrina was investigated by a wave-loading model. The Volume of Fluid (VOF) method was adopted in the model to track the variations of water surface levels. In order to obtain wave parameters and storm-surge elevation at the bridge site during Hurricane Katrina, a storm surge model and a wave propagation model were coupled to hindcast the hydrodynamic conditions. Outputs of the coupled wave-surge models were imported to the wave-loading model to simulate the dynamic wave forces acting on the bridge deck. In order to evaluate the maximum uplift wave force, five different bridge deck elevations submerged at different water depths were investigated. The processes of wave-bridge interaction were simulated by the wave-loading model. The wave profiles, velocity field in the vicinity of the bridge, and dynamic wave forces on the decks were analyzed. Results indicate that the uplift force on the submerged bridge deck span exceeded its own weight under the extreme wave and storm surge conditions during Hurricane Katrina. Moreover, the numerical simulations suggest that the maximum uplift wave force occurred when the storm surge water level reached the top of the bridge deck.     

A Survey of Ocean Simulation and Rendering Techniques in Computer Graphics

This paper presents a survey of ocean simulation and rendering methods in computer graphics. To model and animate the ocean’s surface, these methods mainly rely on two main approaches: on the one hand, those which approximate ocean dynamics with parametric, spectral or hybrid models and use empirical laws from oceanographic research. We will see that this type of methods essentially allows the simulation of ocean scenes in the deep water domain, without breaking waves. On the other hand, physically‐based methods use Navier–Stokes equations to represent breaking waves and more generally ocean surface near the shore. We also describe ocean rendering methods in computer graphics, with a special interest in the simulation of phenomena such as foam and spray, and light’s interaction with the ocean surface.     

A multiphase compressible model for the simulation of multiphase flows

A compressible model able to manage incompressible two-phase flows as well as compressible motions is proposed. After a presentation of the multiphase compressible concept, the new model and related numerical methods are detailed on fixed structured grids. The presented model is a 1-fluid model with a reformulated mass conservation equation which takes into account the effects of compressibility. The coupling between pressure and flow velocity is ensured by introducing mass conservation terms in the momentum and energy equations. The numerical model is then validated with four test cases involving the compression of an air bubble by water, the liquid injection in a closed cavity filled with air, a bubble subjected to an ultrasound field and finally the oscillations of a deformed air bubble in melted steel. The numerical results are compared with analytical results and convergence orders in space are provided.     

Estimated surface-wave contributions to radar Doppler velocity measurements of the ocean surface

For simulated ocean conditions, we estimate the magnitude of the Doppler velocity contributions produced by unresolved surface waves that typical spaceborne synthetic aperture radars (SAR) would measure. The mechanism for generating Doppler velocities is the correlation between wave phase and radar cross section. The contributions analyzed include those of linear gravity waves, second-order wave-wave interactions, Bragg-wave scatterers and breaking waves. For gravity waves, we consider both wave tilt and hydrodynamic modulation transfer functions (MTFs). We find that for nominal sea conditions, the Doppler velocity is significant, on the order of 1 m/s, and exhibits large variation as a function of incidence angle and look with respect to the sea direction. The most important contributors are gravity waves and the Bragg scatterers, followed by sea spikes. Effects produced by second-order wave solutions are argued to be inconsequential.     

Prediction of penetration depth in a plunging water jet using soft computing approaches

The flow characteristics of the plunging water jets can be defined as volumetric air entrainment rate, bubble penetration depth, and oxygen transfer efficiency. In this study, the bubble penetration depth is evaluated based on four major parameters that describe air entrainment at the plunge point: the nozzle diameter (D _N), jet length (L _j), jet velocity (V _N), and jet impact angle (θ). This study presents artificial neural network (ANN) and genetic expression programming (GEP) model, which is an extension to genetic programming, as an alternative approach to modeling of the bubble penetration depth by plunging water jets. A new formulation for prediction of penetration depth in a plunging water jets is developed using GEP. The GEP-based formulation and ANN approach are compared with experimental results, multiple linear/nonlinear regressions, and other equations. The results have shown that the both ANN and GEP are found to be able to learn the relation between the bubble penetration depth and basic water jet properties. Additionally, sensitivity analysis is performed for ANN, and it is found that D _N is the most effective parameter on the bubble penetration depth.     

Pressure bubbles stabilization features in the Stokes problem

The standard finite element approximation using equal-order-linear-continuous velocity–pressure variables is enriched with velocity and pressure bubble functions to model the Stokes problem. We show by static condensation that these bubble functions give rise to a stabilized method involving least-squares forms of the momentum and of the continuity equations. In particular, pressure bubbles play a key role in explaining the addition of the least-squares form of the continuity equation in a stabilized method for Stokes.