Hybrid <Emphasis Type="Italic">N</Emphasis>-order Lagrangian interpolation Eulerian-Lagrangian method for salinity calculation

The Eulerian-Lagrangian method (ELM) has been used by many ocean models as the solution of the advection equation, but the numerical error caused by interpolation imposes restriction on its accuracy. In the present study, hybrid N-order Lagrangian interpolation ELM (LiELM) is put forward in which the N-order Lagrangian interpolation is used at first, then the lower order Lagrangian interpolation is applied in the points where the interpolation results are abnormally higher or lower. The calculation results of a step-shaped salinity advection model are analyzed, which show that higher order (N=3-8) LiELM can reduce the mean numerical error of salinity calculation, but the numerical oscillation error is still significant. Even number order LiELM makes larger numerical oscillation error than its adjacent odd number order LiELM. Hybrid N-order LiELM can remove numerical oscillation, and it significantly reduces the mean numerical error when N is even and the current is in fixed direction, while it makes less effect on mean numerical error when N is odd or the current direction changes periodically. Hybrid odd number order LiELM makes less mean numerical error than its adjacent even number order LiELM when the current is in the fixed direction, while the mean numerical error decreases as N increases when the current direction changes periodically, so odd number of N may be better for application. Among various types of Hybrid N-order LiELM, the scheme reducing N-order directly to 1st-order may be the optimal for synthetic selection of accuracy and computational efficiency.     

Least Square Finite Element Method for Viscous Splitting of Unsteady Incompressible Navier–Stokes Equations

In order to solve unsteady incompressible Navier–Stokes(N–S) equations, a new stabilized finite element method,called the viscous-splitting least square FEM, is proposed. In the model, the N–S equations are split into diffusive and convective parts in each time step. The diffusive part is discretized by the backward difference method in time and discretized by the standard Galerkin method in space. The convective part is a first-order nonlinear equation.After the linearization of the nonlinear part by Newton's method, the convective part is also discretized by the backward difference method in time and discretized by least square scheme in space. C~0-type element can be used for interpolation of the velocity and pressure in the present model. Driven cavity flow and flow past a circular cylinder are conducted to validate the present model. Numerical results agree with previous numerical results, and the model has high accuracy and can be used to simulate problems with complex geometry.     

A Finite Element Solution of Wave Forces on Submerged Horizontal Circular Cylinders

In this paper, a numerical model is established for estimating the wave forces on a submerged horizontal circular cylinder. For predicting the wave motion, a set of two-dimensional Navier-Stokes equations is solved numerically with a finite element method. In order to track the moving non-linear wave surface boundary, the Navier-Stokes equations are discretized in a moving mesh system. After each computational time step, the mesh is modified according to the changed wave surface boundary. In order to stabilize the numerical procedure, a three-step finite element method is applied in the time integration. The water sloshing in a tank and wave propagation over a submerged bar are simulated for the first time to validate the present model. The computational results agree well with the analytical solution and the experimental data.Finally, the model is applied to the simulation of interaction between waves and a submerged horizontal circular cylinder.The effects of the KC number and the cylinder depth on the wave forces are studied.     

Numerical computations of resonant sloshing using the modified isoAdvector method and the buoyancy-modified turbulence closure model

Sloshing is an interfacial-flow phenomenon which brings two challenges on how to locate the position of the interface and avoid the unphysical motion of the interface. In order to locate the the position of the interface, a new geometric Volume-of-Fluid (VOF) method called isoAdvector is adopted to pursue a sharp interface. Aiming to make the isoAdvector method compatible with the dynamic mesh adopted to handle the tank motion, the motion-flux correction is introduced, and a moving-velocity correction for face-interface intersection line (FIIL) is proposed. An approximation formula is adopted to effectively reconstruct the moving-velocity field of the meshes at each cell center based on the motion fluxes on each cell face. In order to avoid the unphysical motion of the interface due to the excessive turbulence level in the transition region at the interface, the buoyancy-modified k − ω SST model is adopted. The numerical results of wave elevations and forces are compared with the experiments. The comparisons suggest that (i) the moving-velocity correction for FIIL is important to update the volume fraction; (ii) the modified isoAdvector method can capture the the position of the interface more accurately than the algebraic VOF method; (iii) the unphysical motion of the interface can be avoided by using the buoyancy-modified k − ω SST model in long-time simulations. In addition, a new post-processing approach is proposed to evaluate the interface thickness. The decrease of interface thickness improves the accuracies of wave elevations by using the modified isoAdvector method. The adoption of both the modified isoAdvector method and the buoyancy-modified k − ω SST model improves the computational accuracies of wave elevations and hydrodynamic loads in long-time simulations.     

Spatial spectra and characteristic horizontal scales of temperature and velocity fluctuations in the convective boundary layer of the atmosphere

Using a high-resolution LES numerical model, we calculated the turbulent thermal convection for high ratios of horizontal and vertical sizes of the computational domain (26: 26: 1). The natural analog of the simulated process is a planetary boundary layer (PBL) of the atmosphere growing with height in the background of stably stratified overlying air layers over a horizontally homogeneous heated surface under a weak average wind. We obtained the spectral distributions of variances of fluctuations in potential temperature and velocity components in ranges corresponding to scales from a few tens of meters to a few tens of kilometers. We found energetically significant segments of the spectrum of large-scale fluctuations in the potential temperature for which the power dependences S ～ k ^?1/3 and S ～ k ^?4/3 are satisfied with good accuracy. We calculated the characteristic spatial scales of horizontal fluctuations in velocity and temperature. We obtained a dependence of these scales on the height of the growing convective PBL. We discuss the characteristic features of large-scale distributions in terms of the self-similarity of the growing boundary layer behavior.     

预修正快速傅里叶变换高阶边界元方法在波浪对物体作用问题中的应用

边界元方法被广泛应用于波浪对海上婕筑物作用领域,但由于传统边界元方法的存储量和计算量均为未知量的平方量级,很难满足大范围多未知量问题的计算需要.本文基于高阶边界元方法,应用预修正快速傅里叶变换方法,使计算量与存储量分别降低至O(NlogN)和O(N)量级,并得到一个连续的压强分布以适应结构设计的要求,同时可以通过使用满...     

A numerical model for onset of scour below offshore pipelines

A numerical model is developed to predict the onset of local scour below offshore pipelines in steady currents and waves. The scour is assumed to start when the pressure gradient underneath the pipeline exceeds the floatation gradient of the sediments. In this model, the water flow field above the bed is determined by solving the two-dimensional (2-D) Reynolds-averaged Navier–Stokes equations with a k-ω turbulence closure. The seepage flow below the seabed is calculated by solving the Darcy's law (Laplace's equation) with known pressure distribution along the common boundaries of the flow domains-seabed. The numerical method used for both the turbulent flow around the pipeline and Darcy's flow in the seabed is a fractional finite element method. The average pressure gradient along the buried pipe surface is employed in the evaluation of onset condition with a calibration coefficient. The numerical model is validated against experimental data available in literature. A unified onset condition for steady currents and waves is proposed. Influences of flow parameters, including water depth, embedment depth, boundary layer thickness, Reynolds number (Re) and Keuleagan–Carpenter (KC) number, on the pressure drop coefficient over the pipeline are studied systematically.     

Experimental validation of a CFD model using a narrow wave flume

Induced swell is characterized in an experimental wave flume that is used to validate the corresponding computational model. The experiments and the numerical simulations are performed in water at several depths (h [m] of 0.2, 0.1, and 0.07), using a piston-type wave maker at set amplitudes (0.015 < A_p [m] < 0.15), accelerations-decelerations (0.3 < a_p < 1.0) and average velocities (0.03 < U [m/s] < 0.3) that control the propagation velocity, the period and the wavelength of the waves. The physical effects are modelled with a 2D computational model (STAR-CCM + v11.02) with a mesh of around 630,000 cells of different adaptive sizes, depending on the region under consideration. The physical model is based on a two-phase Eulerian “Volume of Fluid” unsteady model, accounting for gravity and surface tension, that characterizes turbulence with a k-ε model. A user-defined function, based on the period and the amplitude of the vertical paddle in the wave maker, describes the cyclic motion of the linear induction motor. Both the experimental and the computational results are analyzed taking the validity limits of various wave theories as a reference (Le Méhauté). As a result, the experiments are classified within the intermediate water depth regime that corresponds to the second-order Stokes’ wave theory. In addition, both the wave propagation velocity and the period are represented as a function of the wavelength and compared with the analytical solutions from the wave theories. The experimental and the computational test campaign yielded results that confirmed the validity of the computational model and that defined the most appropriate conditions for a high-quality CFD simulation.     

A simple two-way coupling method of BEM and VOF model for random wave calculations

A numerical method, which combines the boundary element method (BEM) and the volume of the fluid method (VOF method), has been presented to solve wave–structure interactions; the intense wave motion at the proximity of the structure is modeled by the VOF method and the rest of the fluid region is modeled by the BEM. The combined method can considerably reduce the time-consuming VOF domain, and thus practically makes it possible to apply the VOF method for random wave calculations, in which long time computations are usually required to obtain statistically meaningful results, and therefore the use of the single-VOF model often becomes prohibitive in terms of computational time and storage memories. A VOF model CADMAS-SURF, which is based on SMAC scheme and had been constructed by a number of VOF researchers in coastal engineering in Japan, is used in the combined BEM–VOF model. The two-way coupling treatment, which enables us to deal with bidirectional wave propagations, which was originally given for the SOLA-VOF model by Yan et al. (2003a) and later improved by Kim et al. (2007), was modified for the SMAC scheme. The coupling treatments are described in detail in the paper. The validity of the combined BEM–VOF model was investigated by comparing the numerical results with the theoretical results for the propagations of Stokes 5th order waves and random waves.     

杭州湾开挖深水航道对潮流影响的数值研究

采用ｘ方向伸展坐标下的二,三维方程,建立了开避或增深深水航道前后的潮流场数值模模式。该模式在航道横向上网格变距,以保证航道横向上有一定量的网格覆盖。在计算中采用二,三维交替进行,既节省大量计算时间,又保证了计算的稳定性。     

Large eddy simulation of stratified mixing in two-dimensional dam-break problem in a rectangular enclosed domain

Mixing in both coastal and deep ocean emerges as one of the important processes that determines the transport of pollutants, sediments and biological species, as well as the details of the global thermohaline circulation. Both the observations, due to their lack in space and time resolution, and most coastal and general circulation models due to inadequate physics, can only provide partial information about oceanic mixing processes. A new class of nonhydrostatic models supplemented with physically based subgrid-scale (SGS) closures, or so-called large eddy simulation (LES), is put forth as another tool of investigation to complement observational and large-scale modeling efforts.However, SGS models have been developed primarily for homogeneous, isotropic flows. Here, four SGS models based on Smagorinsky eddy viscosity and diffusivity are tested for stratified flows in the context of 2D dam-break problem in a rectangular enclosed domain. This idealized testbed leads to a number of simplifications about the initial conditions, boundary conditions and geometry, while exhibiting the dynamically complex characteristics of stratified flows involving the interaction of shear-induced mixing and internal waves. Direct numerical simulations (DNS) at high resolutions are taken as benchmark solutions. Under-resolved simulations without SGS terms (so-called DNS¹) are used to quantify the impact of SGS stresses. The performance of LES is assessed by using the time evolution of the volume fraction of intermediate density water masses generated by mixing. The simulations are conducted using a nonhydrostatic high-order spectral element model Nek5000 developed to exhibit minimal numerical dissipation and dispersion errors, which is advantageous to quantify accurately the impact of SGS stresses.It is found that all tested SGS models lead to improved results with respect to those from DNS¹. Also, SGS models allow for simulations with coarse resolutions that blow up in DNS¹ due to lack of adequate dissipation where needed. The SGS model in which the vertical eddy diffusion is modulated via a function that depends on the Richardson number Ri shows the most faithful reproduction of mixed water masses at all resolutions tested.The sensitivity of the results to the tunable parameter of the SGS model, to changes in the Ri-dependent function and resolution of the turbulent overturning scales is shown.     

Diagnostic investigation of impulsive pressures induced by plunging breakers impinging on gravel beaches

This paper presents an integrated investigation of physical processes generating impulsive pressures under the action of plunging breakers impinging on gravel beaches. This work is an extension of a recent investigation which suggested that wave impacts from plunging breakers acting on gravel beaches may be a key mechanism to enhance sediment mobilisation. In particular, comparisons of full scale laboratory measurements against model results from a well-validated phase/depth-resolving numerical model based on the Reynolds–Averaged Navier–Stokes (RANS) equations are presented. This represents the first attempt at comparison with such a tool against observed hydrodynamics on steep (slope~1/8) gravel beaches at prototype scale. In order to understand how impulsive pressures are generated under plunging waves, the numerical model is used to carry out a detailed investigation to examine the role of each of the acceleration terms in the momentum balance. Consistent with prior studies, numerical results show that under plunging breakers the local acceleration (∂u/∂t) alone cannot be used as a proxy for pressure gradients. However, the contribution of the third term (w∂u/∂z) of total acceleration is recognized for the first time and indicates that this term has an important role in both the induced pressure gradient and sediment mobilisation as induced by this particular type of breaking. Furthermore, results suggest that a parameterisation of the pressure gradient in terms of ∂u/∂t+u∂u/∂x, may not suffice when dealing with plunging breakers and hence there is a lack of a suitable parameterisation of this process in the present literature. Thus, for different types of breaking it may be necessary to consider a different characterisation of the pressure gradient toward the parameterisation of sediment transport inside the surf zone.     

基于高阶边界元的三维数值波浪港池

初步建立了一个基于高阶边界元的三维数值波浪港池,港池具有造波和消波功能。采用高阶边界元16节点四边形单元和基于二阶显式泰勒展开的混合欧拉-拉格朗日时间步进求解带自由表面的完全非线性势流方程。模型中对于影响数值精度的问题作了细致的处理。数值计算结果表明本港池可以用来模拟非线性波浪的传播,具有很高的数值精度和稳定性。     

Steady streaming around a circular cylinder in an oscillatory flow

Steady streaming induced by an oscillatory flow around a circular cylinder is investigated using a numerical method. Two-dimensional Reynolds-averaged Navier-Stokes equations are solved using a finite element method with a k-ω turbulent model closure. The range of the Keulegan-Carpenter (KC) number investigated is between 2 and 40, which is substantially higher than those reported in literature related to steady streaming to date. A constant value of Stokes number (β) of 196 is chosen in this study. The steady streaming structures and velocity distribution are analysed in detail. It is found that the characteristics of steady streaming are strongly related to the vortex shedding flow regimes.     

Numerical study on the effect of submerged depth on the horizontal plate wave energy converter

The growing search for clean and renewable energy sources has given rise to the studies of exploring sea wave energy. This paper is concerned with the numerical evaluation of the main operational principle of a submerged plate employed for the conversion of wave energy into electrical one. The numerical model used to solve the conservation equations of mass, momentum and transport of volume fraction is based on the finite volume method （FVM）. In order to tackle with the flow of mixture of air-water and its interaction with the device, the multiphase model volume of fluid （VOF） is employed. The purpose of this study is the evaluation of a numerical model for improvement of the knowledge about the submerged plate wave energy converter, as well as the investigation of the effect of the distance from the plate to the bottom of the sea （HP） on the performance of the converter. The simulations for several distances of the plate from the seabed show that the optimal efficiency is 64%, which is obtained for HP=0.53 m （88% of the depth）. This efficiency is 17% larger than that found in the worst case （HP=0.46 m, 77% of the depth）.     

Numerical modeling of flow and scour below a pipeline in currents: Part II. Scour simulation

A vertical two-dimensional (2D) numerical model for time dependent local scour below offshore pipelines subject to unidirectional steady flow is developed. The governing equations for the flow and sediment transport are solved by using finite difference method in a general curvilinear coordinate system. The performance of two turbulence models, the standard k–ɛ model and Smagorinsky subgrid scale (SGS) model, on modeling time dependent scour processes is examined. Both suspended load and bed load are considered in the scour model. The suspended-load model is verified against two channel sediment transport cases. The change of bed level is calculated from the continuity equation of total sediment transport. A new time marching scheme and a sand slide scheme are proposed for the scour calculation. It is found that the proposed time marching scheme and sand slide model work well for both clear-water and live-bed scour situations and the standard k–ɛ turbulence closure is more preferable than the SGS model in the 2D scour model developed in this study.     

Buckling of cross-ply cylinders under hydrostatic pressure considering pressure stiffness

Buckling behavior of cross-ply cylinders under hydrostatic pressure is investigated using a semi-analytical finite element based on a consistent first order shear deformable shell theory. Potential loss due to external pressure, also called pressure stiffness (PS) is taken into account by making use of Koiter's related energy expression. A number of verification problems are solved and the numerical results are compared with the analytical results available in the literature and excellent agreement is observed. New numerical results are presented to assess the effect of PS on buckling due to hydrostatic pressure. It is shown that PS causes a decrease in the buckling load and this decrease depends on the size of the cylinder and the material. Also, issues related to thickness optimization are examined and optimal lamina thicknesses are determined for a number of cases with and without PS taken into account.     

Free-surface evolution and wave kinematics for nonlinear uni-directional focused wave groups

This paper concerns the propagation of transient wave groups, focused at a point in time and space to produce locally large waves having a range of steepness. The experimental study was carried out in a wave flume at Dalian University of Technology. The numerical simulations were based on a nonlinear boundary integral equation solved by a higher-order boundary element method (HOBEM). Rather than simulate the whole experimental tank, local surface elevation measurements were used to drive the numerical solution from a point less than two wavelengths upstream of the focus position, leading to significant savings in computational time. Excellent agreement is achieved between the water surface elevations and the water particle kinematics measured in the experiments and those predicted numerically at wave group focus, even for near-breaking waves up to a steepness of kA=0.405 for which even locally matched 2nd-order theory is inadequate. Results based on the linear and 2nd-order theory are also presented in the comparisons. When compared with the first- and 2nd-order solutions, the fully nonlinear wave–wave interactions produce a steeper wave envelope in which the central wave crest is higher and narrower, while the adjacent wave troughs are broader and less deep.     

Numerical modeling of nonlinear resonance of semi-enclosed water bodies: Description and experimental validation

The purpose of this work is to a present a numerical model to solve a set of modified Boussinesq equations to analyse nonlinear resonance of semi-enclosed water bodies. The equations are solved on a finite element unstructured grid in order to achieve an optimal mesh resolution with the local geometry. The model is able to simulate long time lapses and realistic forcing in real bathymetries with a reasonable computational cost. To validate the numerical results, a set of experiments was carried out in a physical model of two adjacent elongated basins. Comparisons between numerical and experimental results for different geometries and nonlinear conditions show that the model is able to simulate with an excellent agreement the transient nonlinear resonant process.