Stephan Guignard

Local and non-local curvature approximation: a new asymptotic theory for wave scattering

Abstract We present a new asymptotic theory for scalar and vector wave scattering from rough surfaces which federates an extended Kirchhoff approximation (EKA), such as the integral equation method (IEM), with the first and second order small slope approximations (SSA). The new development stems from the fact that any improvement of the ‘high frequency’ Kirchhoff or tangent plane approximation (KA) must come through surface curvature and higher order derivatives. Hence, this condition requires that the second order kernel be quadratic in its lowest order with respect to its Fourier variable or formally the gradient operator. A second important constraint which must be met is that both the Kirchhoff approximation (KA) and the first order small perturbation method (SPM-1 or Bragg) be dynamically reached, depending on the surface conditions. We derive herein this new kernel from a formal inclusion of the derivative operator in the difference between the polarization coefficients of KA and SPM-1. This new kernel is as simple as the expressions for both Kirchhoff and SPM-1 coefficients. This formal difference has the same curvature order as SSA-1 + SSA-2. It is acknowledged that even though the second order small perturbation method (SPM-2) is not enforced, as opposed to the SSA, our model should reproduce a reasonable approximation of the SPM-2 function at least up to the curvature or quadratic order. We provide three different versions of this new asymptotic theory under the local, non-local, and weighted curvature approximations. Each of these three models is demonstrated to be tilt invariant through first order in the tilting vector.

Solitary wave breaking on sloping beaches: 2-D two phase flow numerical simulation by SL-VOF method

Abstract This paper describes the development of a computational method for simulating breaking and post-breaking of solitary waves over sloping bottoms. The Navier–Stokes equations are solved in air and water with respect to the real density ratio between the two fluids using a pseudo-compressibility method. The interface tracking is achieved by a new method, called Segment Lagrangian-Volume Of Fluid (SL-VOF), using the well known concepts of VOF, Piecewise Linear Interface Calculation (PLIC) and adding Lagrangian advection of the segments representing the interface. The verification of this method is made through various simple test cases. Results concerning wave shoaling are compared with those of Boundary Integral Element Method (BIEM) simulations, that are known to be very accurate up to breaking. The SL-VOF method is able to simulate the flow beyond the point at which the interface impacts on itself. Simulations of the breaking stage are compared with experiments. In both cases a very good agreement is observed.

Formal tilt invariance of the local curvature approximation

Abstract Tilt invariance is a stringent but necessary condition that a second-order wave scattering model must satisfy in order to qualify for a broad range of applications. This invariance expresses the fact that the scattering model is unchanged whether the tilting of the scattering surface is implemented before or after its reduction to the limit of the small-perturbation method (SPM). Our scattering model is based on a second-order kernel which is quadratic in its lowest order with respect to successive derivatives of the rough surface. Hence, it is termed the local curvature approximation (LCA). We have previously demonstrated that the LCA is approximately tilt invariant in the quasi-specular and quasi-backscattering geometries. In this contribution, LCA is made formally tilt invariant up to first order in the tilting vector. It will be shown that this formal tilt invariance is achieved mainly through inclusion of polarization mixing due to out-of-plane tilt. Even though the LCA formally reduces to the SPM and Kirchhoff limits in addition to tilt invariance, its curvature kernel stays reasonably concise and practical to implement in both analytical and numerical evaluations. This curvature kernel may also be used in the other two formulations of our model, namely the non-local curvature approximation and the weighted curvature approximation.

A practical second-order electromagnetic model in the quasi-specular regime based on the curvature of a ‘good-conducting’ scattering surface

Abstract This letter presents an approximate second-order electromagnetic model where polarization coefficients are surface dependent up to the curvature order in the quasi-specular regime. The scattering surface is considered ‘good-conducting’ as opposed to the case for our previous derivation where perfect conductivity was assumed. The model reproduces dynamically, depending on the properties of the scattering surface, the tangent-plane (Kirchhoff) or the first-order small-perturbation (Bragg) limits. The convergence is assumed to be ensured by the surface curvature alone. This second-order model is shown to be consistent with the small-slope approximation of Voronovich (SSA-1+SSA-2) for perfectly conducting surfaces. Our model differs from SSA-1 + SSA-2 in its dielectric expression, to correct for a full convergence toward the tangent-plane limit under the ‘good-conducting’ approximation. This new second-order formulation is simple because it involves a single integral over the scattering surface and therefore it is suitable for a vast array of analytical and numerical applications in quasi-specular applications.

Formal tilt invariance of the nonlocal curvature approximation and its connection to the integral equation method

Tilt invariance is a stringent constraint that second-order scattering models such as the integral equation method (IEM) should satisfy in order to expand their domain of applicability. Moreover, second-order scattering models must reproduce elementary limits such as the small perturbation method (SPM) and the high-frequency Kirchhoff approximation. Tilt invariance is met if and only if a scattering model yields the same asymptotic limit whether the scattering surface is tilted before or after the limiting process. In particular, the tilted SPM coefficients are well determined by simply tilting the reference frame. If it is tilt invariant, a second-order scattering model will reproduce these tilted coefficients by simply tilting the surface explicitly present in the expression of the scattering model before reducing it to the SPM limit. In this letter, we demonstrate that our nonlocal curvature approximation (NLCA) is formally tilt invariant up to first order in the tilting vector. Satisfying the tilt invariance property can extend the applicability of scattering models to account, for example, for scattering from multiscale surfaces and polarization mixing due to out-of-plane tilting. It is also suggested that replacing the field coefficients of IEM by the curvature kernel of NLCA introduces a promising alternative technique that includes multiple scattering up to double reflections from the rough surface, while remaining analytically compact and formally tilt invariant.

Implementation and Validation of a Breaker Model in a Fully Nonlinear Wave Propagation Model

A spilling breaker model is implemented in a two-dimensional fully nonlinear coastal wave propagation model. A maximum surface slope breaking criterion is used to identify breaking waves within the incident wave train. Energy dissipation is achieved by specifying an absorbing surface pressure over breaking wave crest areas. The pressure is proportional to the normal particle velocity on the free surface. The instantaneous power dissipated in each breaking wave is speci ed proportional to the dissipation in a hydraulic jump of identical characteristics. Computations for a periodic wave shoaling and breaking over a plane slope are compared to laboratory experiments. The agreement is quite good, although more work remains to be done in re ning the breaker model parameters.

Two-dimensional front-tracking model for film evaporation

To understand the physical process involved in film evaporation, a new numerical model is created using coupled quadratic finite element formulation of the conservation equations. The heat transport equation is solved in the three different phases (solid, liquid and vapor) while the Navier-Stokes equation are solved in the two fluids. The gradient discontinuity at the liquid vapor interface provides local value of the evaporative flux density that is directly linked to the interface velocity jump through mass conservation principle and used as boundary condition for two fluid flow computations. Testing on academic cases and application to axisymmetric film evaporation including comparison with experiments are shown.

A time-frequency application with the Stokes-Woodward technique

Elfouhaily et al. (2003) generalized Woodwards theorem and applied it to the case of random signals jointly modulated in amplitude and frequency. This generalization yields a new spectral technique to estimate the amount of energy due to mode coupling without calling for higher order statistics. Two power spectra are detected; the first is related to the independent modes, and the second contains extra energy caused by mode coupling. This detection is now extended from frequency to time-frequency domain. A comparison between a wavelet transform and our time-frequency technique shows good agreement along with new insight into the time occurrence of the nonlinearities or mode coupling. An application to water surface waves is given as an example.