Numerical simulation of oblique and multidirectional wave propagation and breaking on steep slope based on FEM model of Boussinesq equations

Creating a representative numerical simulation of the propagation and breaking of waves along slopes is an important problem in engineering design. Most studies on wave breaking have focused on the propagation of normal incident waves on gentle slopes. In practice, however, waves on steep slopes are obliquely incident or multidirectional irregular waves. In this paper, the eddy viscosity term is introduced to the momentum equation of the improved Boussinesq equations to model wave dissipation caused by breaking and friction, and a numerical model based on an unstructured finite element method (FEM) is established based on the governing equations. It is applied to simulate wave propagation on a steep slope of 1:5. Parallel physical experiments are conducted for comparative analysis that considered a large number of cases, including those featuring of normal and oblique incident regular and irregular waves, and multidirectional waves. The heights of the incident wave increase for different periods to represent different kinds of waves breaking. Based on examination, the effectiveness and accuracy of the numerical model is verified through a comprehensive comparison between the numerical and the experimental results, including in terms of variation in wave height, wave spectrum, and nonlinear parameters. Satisfactory agreement between the numerical and experimental values shows that the proposed model is effective in representing the breaking of oblique incident regular waves, irregular waves, and multidirectional incident irregular waves. However, the initial threshold of the breaking parameter η_t^(I) takes different values for oblique and multidirectional waves. This needs to be paid attention when the breaking of waves is simulated using the Boussinesq equations.     

Two-phase flow modelling of spilling and plunging breaking waves

A two-phase flow model, which solves the flow in the air and water simultaneously, has been employed to investigate both spilling and plunging breakers in the surf zone with a focus during wave breaking. The model is based on the Reynolds-averaged Navier–Stokes equations with the k–?$k''–''?$ turbulence model. The governing equations are solved using the finite volume method, with the partial cell treatment being implemented in a staggered Cartesian grid to deal with complex geometries. The PISO algorithm is utilised for the pressure–velocity coupling and the air–water interface is modelled by the interface capturing method via a high-resolution volume of fluid scheme. Numerical results are compared with experimental measurements and other numerical studies in terms of water surface elevations, mean flow and turbulence intensity, in which satisfactory agreement is obtained. In addition, water surface profiles, velocity and vorticity fields during wave breaking are also presented and discussed. It is shown that the present model is capable of simulating the wave overturning, air entrainment and splash-up processes.     

Modelling of wave phenomena in the Zahorski material based on modified library for ADINA software

This paper presents numerical modelling of wave phenomena in simple elastic structures such as rods and shields made of hyperelastic Zahorski material. The main difference between the Zahorski material, which is an elastic material in the Green sense, and the commonly used Mooney–Rivlin material lies in the non-linear term including the constant C₃. Consequently, qualitative and quantitative differences are observed compared to the Mooney–Rivlin material, for example in the values of effective stresses. The extension to the ADINA software developed by the author, which helps create 2D and 3D libraries, significantly facilitates modelling of the Zahorski material. The modification can be used for comparison of wave phenomena that are observed during the propagation of disturbances in the Mooney–Rivlin and Zahorski materials. It should be emphasised that the Zahorski material behaves much better at high strains during the analysis of incompressible rubber and rubber-like hyperelastic materials and can be used in various fields of science wherever the model of Mooney–Rivlin material is successfully applied. The results of numerical computations for both Mooney–Rivlin and Zahorski materials were presented in a graphical form and compared in order to illustrate the differences.     

桩基承台结构的波流力研究进展

我国近海风电场建设大多采用桩基承台结构．总结了不规则波浪和水流共同作用下桩基承台结构的波流力物理模型实验结果,得到了群桩效应系数及其变化规律,讨论了作用于近水面承台底部的波浪拍击力;从理论上分析了规则波作用下承台对桩基波浪力的影响;建立了规则波与桩基承台相互作用的数值模型,揭示了波浪在承台的上浪与爬高及其水动力特征．鉴于桩基承台结构包含多个斜桩和较大尺度的承台,在波浪与水流作用下该结构物附近的流场结构十分复杂,有必要针对结构附近的流动结构以及自由表面大变形开展细致的实验和数值模拟研究,以进一步揭示作用于这类结构的波流力变化规律及其机理.     

Study of wave–wind interaction at a seawall using a numerical wave channel

This paper presents the study on wind and waves interactions at a seawall using a numerical wave channel. The numerical experiments were conducted for wave overtopping of a 1/4 sloping seawall using several conditions of incident waves and wind speeds. The numerical results were verified against laboratory data in a case for wave overtopping without wind effects. The interaction of waves and wind was analyzed in term of mean wave quantities, overtopping rate and variation of wind velocity at some selected locations. The results showed that the overtopping rate was strongly affected by wind and the wind field was also significantly modified by waves. There exists an effective range of wind speed in comparison with the local shallow wave speed at the breaking location, which gives significant effects to the wave overtopping rates. The maximum of wind adjustment coefficient f_w for wave overtopping rate was strongly related to the mean overtopping rate in the case for no wind. This study also showed that when the mean overtopping rate was greater than 5 × 10⁻⁴ m³/s/m, the maximum of wind adjustment coefficient f_w approached to a specific value of about 1.25.     

DNS of stochastically forced laminar plane Couette flow

Background of three dimensional hydrodynamic/vortical fluctuations in a stochastically forced, laminar, incompressible, plane Couette flow is simulated by direct numerical simulations (DNS). It was found that the fluctuating field has well pronounced peculiarities: (i) The hydrodynamic fluctuations exhibits non–exponential, transient growth; (ii) Streamwise non–constant fluctuations with the characteristic length scale of the order of the channel width are predominant in the fluctuating spectrum; (iii) Existence of coherent structures in the fluctuating background; (iv) Stochastic forcing breaks the spanwise reflection symmetry (inherent to the linear and full Navier–stokes equations in a case of the Couette flow) and inputs an asymmetry on dynamical processes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical study on viscoelastic fluid flow past a rigid body

Viscoelastic non-Newtonian fluids can be achieved by adding a small amount of polymer additives to a Newtonian fluid. In this paper, numerical simulations are used to investigate the influence of such polymer additives on the behavior of flow past a circular cylinder. A numerical method is proposed that discretizes the non-linear viscoelastic system on a uniform Cartesian grid, with a penalization method to model the presence of the cylinder. The drag of the cylinder and the flow behavior under the effect of different Reynolds numbers ($\mathrm{Re}$), Weissenberg numbers (Wi) and polymer viscosity ratios (ε) are studied. Numerical results show that different flow characteristics are exhibited in different parameter zones. The polymer viscosity ratio plays an important role at low Weissenberg and Reynolds numbers, but as the Reynolds and Weissenberg numbers increase, the influence of ε weakens. The drag force of the cylinder is mostly affected by the Reynolds and Weissenberg numbers. At low Reynolds numbers, the drag of the cylinder and the flow fields are only affected by a large value of Wi when the elastic forces are strong. Non-trivial drag reduction occurs only when there is vortex shedding in the wake flow, whereas drag enhancement happens when the vortex shedding is inhibited.     

Spatio-Temporal Dynamical Systems in Inner-Shell Photoionization in Free Molecules,Clusters, and Solids

Dynamic spatial hole localization and symmetry breaking phenomena are examined. Absorption of X-ray synchrotron and free-electron–laser radiation in matter is accompanied by strong dynamic corehole localization and temporary trap of the electron ejected from a deep level within the finite size potential barrier. As a result the symmetry of core excited states is reduced in comparison with ground state as the inversion symmetry is being broken. This is a very general property of coreexcited polyatomic compounds with equivalent atoms as their equivalence implies their equal probability of excitation averaged over large timescale but not simultaneous core excitation. Different approaches to rationalizing the symmetry breaking phenomena are presented and discussed with the emphasis on the quasiatomic dynamic corehole localization model. By examining the experimental ultrafast probe of photoabsorption processes we demonstrate an important role of spatio-temporal (nanometric-femtosecond) dynamically localized coreexcited moieties in molecule, clusters and solids. The photoelectron angular distributions from N and O 1s levels in fixed-in-space N₂ and CO₂ molecules, the photoelectron induced rotational heating of N₂, the Auger decay spectra of N₂ and the near S 1s edge X-ray absorption fine structure of free SF₆ molecules are discussed in more detail.     

Coherent structure in flow over a slitted bluff body

The focus of the present investigation is resolution of the coherent structure in the near wake behind a slitted bluff body. The bluff body is two-dimensional with gap ratio from 0.12 to 0.48. The evolution of the structure was numerically investigated using the renormalization group (RNG) k–ε model at Reynolds number of 470,000. Two types of coherent structure are identified: At low gap ratio 0.12, the structure is characterized by a flip–flopping gap flow; at high ratio 0.22–0.48, the gap flow deflects to one side with an asymmetrical wake. The coherent structure is divided by the gap flow into two zones called the primary recirculation zone and the secondary recirculation zone. The coherent structure is intimately related to the gap ratio, and the structure of small gap ratio is different from that of large gap ratio because the interaction between two zones relates to the gap ratio. To explain the vortex shedding, a mechanism that single vortex of large size suddenly immerses between two shear layers was proposed. Experimental results using point-to-point method and particle-image velocimetry (PIV) measurements in a close wind tunnel were also carried out to confirm the observation from the numerical study. The evidence shows that the numerical results are of good agreement with the experiments. The comparison between the RNG k–ε model and the large eddy simulation also indicates that the RNG k–ε model is adequate in computing the bluff body flow.     

Simplification des cartes géographiques par minimisation de la déformation localeTerrain simplification by minimization of the local deformation

In cartography, the geographic regions are usually represented using regular dense maps corresponding to heights values associated with the nodes of a regular grid of R². The simplification of such maps is an absolute requirement in order to make storage, simulation and display possible. In this Note, we propose a new simplification method based on a measure of the local deformation of the surface. The latter allows, in particular, minimization of the approximation error during the simplification. A numerical example is provided to emphasize the efficiency of this approach. To cite this article: P.J. Frey, H. Borouchaki, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 227–232.     

Adaptive FEM–BEM coupling with a Schur complement error indicator

In this paper an a-posteriori error estimate for non-linear coupled FEM–BEM equations is derived by using the Steklov–Poincaré operator and hierarchical basis techniques. We obtain “local” error indicators which are based on two-level subspace decompositions with the additive Schwarz operator. We present an algorithm for adaptive error control which is driven by these error indicators and numerical results are included.     

Hydrodynamic analogy for direct shear force and bending moment evaluation in a plate

A brand new interpretation of the plate bending equations is given using hydrodynamic analogy. It permits one to determine directly the shear forces and bending moments of a plate without the need of finding deflections. In engineering design of a plate it is more important to know shear forces and bending moments than the deflections. The existing numerical methods of solution of plate problems consist in determining deflections; then shear forces and bending moments are obtained by differentiating the deflection three and two times which produces great loss of accuracy. The hydrodynamic analogy method has the advantage over other numerical methods because the shear forces and bending moments are obtained directly, without the need of finding deflections and because they are obtained with better accuracy. The hydrodynamic analogy can be applied to a plate of arbitrary shape, with arbitrary boundary conditions under an arbitrary loading.     

Wave breaking in solutions of the dispersionless kadomtsev-petviashvili equation at a finite time

We discuss some interesting aspects of the wave breaking in localized solutions of the dispersionless Kadomtsev-Petviashvili equation, an integrable partial differential equation describing the propagation of weakly nonlinear, quasi-one-dimensional waves in 2+1 dimensions, which arise in several physical contexts such as acoustics, plasma physics, and hydrodynamics. For this, we use an inverse spectral transform for multidimensional vector fields that we recently developed and, in particular, the associated inverse problem, a nonlinear Riemann-Hilbert problem on the real axis. In particular, we discuss how the derivative of the solution blows up at the first breaking point in any direction of the plane (x, y) except in the transverse breaking direction and how the solution becomes three-valued in a compact region of the plane (x, y) after the wave breaking.     

High accuracy cubic spline finite difference approximation for the solution of one-space dimensional non-linear wave equations

In this paper, we propose a new three-level implicit nine point compact cubic spline finite difference formulation of order two in time and four in space directions, based on cubic spline approximation in x-direction and finite difference approximation in t-direction for the numerical solution of one-space dimensional second order non-linear hyperbolic partial differential equations. We describe the mathematical formulation procedure in details and also discuss how our formulation is able to handle wave equation in polar coordinates. The proposed method when applied to a linear hyperbolic equation is also shown to be unconditionally stable. Numerical results are provided to justify the usefulness of the proposed method.     

Series solutions and a perturbation formula for the extended Rayleigh problem of hydrodynamic stability

We generalize Tollmien’s solutions of the Rayleigh problem of hydrodynamic stability to the case of arbitrary channel cross sections, known as the extended Rayleigh problem. We prove the existence of a neutrally stable eigensolution with wave number k _s ?>?0; it is also shown that instability is possible only for 0?<?k?<?k _s and not for k?>?k _s . Then we generalize the Tollmien–Lin perturbation formula for the behavior of c _i, the imaginary part of the phase velocity as the wave number k →k _s ? to the extended Rayleigh problem and subsequently, we use this formula to demonstrate the instability of a particular shear flow.     

Modeling cadmium accumulation in radish,carrot, spinach and cabbage

Heavy metals like cadmium and arsenic have serious health consequences and ecosystem impacts. Due to various factors including the disposal of municipal and industrial wastes, application of fertilizers, atmospheric deposition and discharge of wastewater on land, has resulted in increase in the concentration of heavy metals in the soil. Crops and vegetables grown on such soil accumulate heavy metals, which leads to phyto-toxicity. For understanding and managing precious natural resources, mathematical models are increasingly being used. This paper describes a dynamic macroscopic numerical model for heavy metal transport and its uptake by vegetables in the root zone. The model is applied for simulating cadmium uptake by radish (Raphanus sativus), carrot (Daucos carota), spinach (Spinacia oleracea) and cabbage (Brassica oleracea) by using measured field data. The governing non-linear partial differential equations are solved numerically by an implicit finite difference method using Picard’s iterative technique and the source code is written in MATLAB.     

Stimulated Raman scattering in the presence of trapped particle instability

The behaviour of electron gas in a laser plasma corona is studied in the presence of stimulated Raman scattering (SRS). The 1D Vlasov–Maxwell model describing plasma relevant to the experiment PALS (Prague Asterix Laser System), where the nanosecond iodine laser with the wavelength of first harmonic λ_vac = 1.3152 and with the power density in the focal spot I = 10²⁰ W/m² is in operation. For the solution of Vlasov equation for the electron distribution function a Fourier–Hermite transform method is used. For the numerical stabilization a small collision term is added to the Vlasov equation keeping its value realistic for the condition relevant to the PALS experiment. The dominant wave modes in our model are both the backward (SRS-B) and the forward (SRS-F) Raman scattering, each of them accompanied by the forward going electron plasma wave. Several mechanisms were identified such as the SRS-B plasma wave spectral broadening due to a trapped particle instability (TPI) or the formation of an electrostatic quasi-mode by non-resonant interaction of SRS-B and SRS-F plasma waves.     

Periodic solutions of non-linear discrete Volterra equations with finite memory

In this paper we discuss the existence of periodic solutions of discrete (and discretized) non-linear Volterra equations with finite memory. The literature contains a number of results on periodic solutions of non-linear Volterra integral equations with finite memory, of a type that arises in biomathematics. The “summation” equations studied here can arise as discrete models in their own right but are (as we demonstrate) of a type that arise from the discretization of such integral equations. Our main results are in two parts: (i) results for discrete equations and (ii) consequences for quadrature methods applied to integral equations. The first set of results are obtained using a variety of fixed-point theorems. The second set of results address the preservation of properties of integral equations on discretizing them. The effect of weak singularities is addressed in a final section. The detail that is presented, which is supplemented using appendices, reflects the differing prerequisites of functional analysis and numerical analysis that contribute to the outcomes.