2D unstructured mesh generation for oceanic and coastal tidal models from a localized truncation error analysis with complex derivatives

A method for computing target element size for tidal, shallow water flow is developed and demonstrated. The method, Localized truncation error analysis with complex derivatives (LTEA-CD) utilizes localized truncation error estimates of the linearized shallow water momentum equations consisting of complex derivative terms. This application of complex derivatives is the chief way in which the method differs from a similar existing method, LTEA. It is shown that LTEA-CD produces results that are essentially equivalent to those of LTEA (which in turn has been demonstrated to be capable of producing practicable target element sizes) with reduced computational cost. Moreover, LTEA-CD is capable of computing truncation error and corresponding target element sizes at locations up to and including the boundary, whereas LTEA can be applied only on the interior of the model domain. We demonstrate the convergence of solutions over meshes generated with LTEA-CD using an idealized representation of the western North Atlantic Ocean, Caribbean Sea and Gulf of Mexico.     

Automatic,unstructured mesh generation for tidal calculations in a large domain

An automated procedure is described for the production of unstructured, finite element meshes to perform depth-integrated, hydrodynamic calculations in an ocean-scale, two-dimensional domain. Three relatively coarse meshes with nearly identical boundaries are automatically produced by basing internal size guidelines on a localized truncation error analysis that was performed using results from a highly resolved mesh.

Qualitative and quantitative comparisons of model performance are made at 150 historical tidal stations. The coarsest mesh is shown to meet or exceed the overall accuracy of the other meshes, including a highly resolved mesh that has over six times as many computational points. The automated procedure quickly and easily produces a computationally efficient and accurate finite element mesh that is reproducible. In addition, the methodology is shown to have potential for assessing the importance and accuracy of and bathymetric details and evaluating historical hydrodynamic data.     

Incorporating spatially variable bottom stress and Coriolis force into 2D,a posteriori,unstructured mesh generation for shallow water models

An enhanced version of our localized truncation error analysis with complex derivatives (LTEA?CD ) a posteriori approach to computing target element sizes for tidal, shallow water flow, LTEA+CD , is applied to the Western North Atlantic Tidal model domain. The LTEA + CD method utilizes localized truncation error estimates of the shallow water momentum equations and builds upon LTEA and LTEA?CD‐based techniques by including: (1) velocity fields from a nonlinear simulation with complete constituent forcing; (2) spatially variable bottom stress; and (3) Coriolis force. Use of complex derivatives in this case results in a simple truncation error expression, and the ability to compute localized truncation errors using difference equations that employ only seven to eight computational points. The compact difference molecules allow the computation of truncation error estimates and target element sizes throughout the domain, including along the boundary; this fact, along with inclusion of locally variable bottom stress and Coriolis force, constitute significant advancements beyond the capabilities of LTEA. The goal of LTEA + CD is to drive the truncation error to a more uniform, domain‐wide value by adjusting element sizes (we apply LTEA + CD by re‐meshing the entire domain, not by moving nodes). We find that LTEA + CD can produce a mesh that is comprised of fewer nodes and elements than an initial high‐resolution mesh while performing as well as the initial mesh when considering the resynthesized tidal signals (elevations). Copyright © 2008 John Wiley & Sons, Ltd.     

One‐dimensional finite element grids based on a localized truncation error analysis

With the exponential increase in computing power, modelers of coastal and oceanic regions are capable of simulating larger domains with increased resolution. Typically, these models use graded meshes wherein the size of the elements can vary by orders of magnitude. However, with notably few exceptions, the graded meshes are generated using criteria that neither optimize placement of the node points nor properly incorporate the physics, as represented by discrete equations, underlying tidal flow and circulation to the mesh generation process. Consequently, the user of the model must heuristically adjust such meshes based on knowledge of local flow and topographical features—a rough and time consuming proposition at best. Herein, a localized truncation error analysis (LTEA) is proposed as a means to efficiently generate meshes that incorporate estimates of flow variables and their derivatives. In a one‐dimensional (1D) setting, three different LTEA‐based finite element grid generation methodologies are examined and compared with two common algorithms: the wavelength to Δx ratio criterion and the topographical length scale criterion. Errors are compared on a per node basis. It is shown that solutions based on LTEA meshes are, in general, more accurate (both locally and globally) and more efficient. In addition, the study shows that the first four terms of the ordered truncation error series are in direct competition and, subsequently, that the leading order term of the truncation error series is not necessarily the dominant term. Analyses and results from this 1D study lay the groundwork for developing an efficient mesh generating algorithm suitable for two‐dimensional (2D) models. Copyright © 2000 John Wiley & Sons, Ltd.     

Two‐dimensional,unstructured mesh generation for tidal models

The successful implementation of a finite element model for computing shallow‐water flow requires the identification and spatial discretization of a surface water region. Since no robust criterion or node spacing routine exists, which incorporates physical characteristics and subsequent responses into the mesh generation process, modelers are left to rely on crude gridding criteria as well as their knowledge of particular domains and their intuition. Two separate methods to generate a finite element mesh are compared for the Gulf of Mexico. A wavelength‐based criterion and an alternative approach, which employs a localized truncation error analysis (LTEA), are presented. Both meshes have roughly the same number of nodes, although the distribution of these nodes is very different. Two‐dimensional depth‐averaged simulations of flow using a linearized form of the generalized wave continuity equation and momentum equations are performed with the LTEA‐based mesh and the wavelength‐to‐gridsize ratio mesh. All simulations are forced with a single tidal constituent, M₂. Use of the LTEA‐based procedure is shown to produce a superior (i.e., less error) two‐dimensional grid because the physics of shallow‐water flow, as represented by discrete equations, are incorporated into the mesh generation process. Copyright © 2001 John Wiley & Sons, Ltd.     

The effect of using non-orthogonal boundary-fitted grids for solving the shallow water equations

The purpose of this paper is to examine the effects of using non-orthogonal boundary-fitted grids for the numerical solution of the shallow water equations. Two geometries with well known analytical solutions are introduced in order to investigate the accuracy of the numerical solutions. The results verify that a reasonable departure from orthogonality can be allowed when the rate of change of cell areas is kept sufficiently small (i.e. it is not necessary to create a strictly orthogonal grid when the grid is sufficiently smooth).     

A class of fully third-order accurate projection methods for solving the incompressible Navier-Stokes equations

In this paper, a fully third-order accurate projection method for solving the incompressible Navier-Stokes equations is proposed. To construct the scheme, a continuous projection procedure is firstly presented. We then derive a sufficient condition for the continuous projection equations to be temporally third-order accurate approximations of the original Navier-Stokes equations by means of the local- truncation-error-analysis technique. The continuous projection equations are discretized temporally and spatially to third-order accuracy on the staggered grids, resulting in a fully third-order discrete projection scheme. The possibility to design higher-order projection methods is thus demonstrated in the present paper. A heuristic stability analysis is performed on this projection method showing the probability of its being stable. The stability of the present scheme is further verified through numerical tests. The third-order accuracy of the present projection method is validated by several numerical test cases. The project supported by the China NKBRSF (2001CB409604) The English text was polished by Yunming Chen     

Method of relaxed streamline upwinding for hyperbolic conservation laws

In this work a new finite element based Method of Relaxed Streamline Upwinding is proposed to solve hyperbolic conservation laws. Formulation of the proposed scheme is based on relaxation system which replaces hyperbolic conservation laws by semi-linear system with stiff source term also called as relaxation term. The advantage of the semi-linear system is that the nonlinearity in the convection term is pushed towards the source term on right hand side which can be handled with ease. Six symmetric discrete velocity models are introduced in two dimensions which symmetrically spread foot of the characteristics in all four quadrants thereby taking information symmetrically from all directions. Proposed scheme gives exact diffusion vectors which are very simple. Moreover, the formulation is easily extendable from scalar to vector conservation laws. Various test cases are solved for Burgers equation (with convex and non-convex flux functions), Euler equations and shallow water equations in one and two dimensions which demonstrate the robustness and accuracy of the proposed scheme. New test cases are proposed for Burgers equation, Euler and shallow water equations. Exact solution is given for two-dimensional Burgers test case which involves normal discontinuity and series of oblique discontinuities. Error analysis of the proposed scheme shows optimal convergence rate. Moreover, spectral stability analysis gives implicit expression of critical time step.     

The influence of normal flow boundary conditions on spurious modes in finite element solutions to the shallow water equations

Finite element solutions of the primitive equation (PE) form of the shallow water equations are notorious for the severe spurious 2Δx modes which appear. Wave equation (WE) solutions do not exhibit these numerical modes. In this paper we show that the severe spurious modes in PE solutions are strongly influenced by essential normal flow boundary conditions in the coupled continuity-momentum system of equations. This is demonstrated through numerical examples that avoid the use of essential normal flow boundary conditions either by specifying elevation values over the entire boundary or by implementing natural flow boundary conditions in the weak weighted residual form of the continuity equation. Results from a series of convergence tests show that PE solutions are of nearly the same quality as WE solutions when spurious modes are suppressed by alternative specification of the boundary conditions. Network intercomparisons indicate that varying nodal support does not excite spurious modes in a solution, although it does enhance the spurious modes introduced when an essential normal flow boundary condition is used. Dispersion analysis of discrete equations for interior and boundary nodes offers an explanation of the observed solution behaviour. For certain PE algorithms a mixed situation can arise where the boundary nodes exhibit a monotonic (noise-free) dispersion relationship and the interior nodes exhibit a folded (noisy) dispersion relationship. We have found that the mixed situation occurs when all boundary nodes are specified elevation nodes (which are enforced as essential conditions in the continuity equation) or when specified flow boundary nodes are treated as natural boundary conditions in the continuity equation. In either case the effect is to generate a solution that is essentially free of noise. Apparently, the monotonic dispersion behaviour at the boundaries suppresses the otherwise noisy behaviour caused by the folded dispersion relation on the interior.     

Analytical and numerical studies for Seiches in a closed basin with bottom friction

A standing wave oscillation in a closed basin, known as a seiche, could cause destruction when its period matches the period of another wave generated by external forces such as wind, quakes, or abrupt changes in atmospheric pressure. It is due to the resonance phenomena that allow waves to have higher amplitude and greater energy, resulting in damages around the area. One condition that might restrict the resonance from occurring is when the bottom friction is present. Therefore, a modified mathematical model based on the shallow water equations will be used in this paper to investigate resonance phenomena in closed basins and to analyze the effects of bottom friction on the phenomena. The study will be conducted for several closed basin shapes. The model will be solved analytically and numerically in order to determine the natural resonant period of the basin, which is the period that can generate a resonance. The computational scheme proposed to solve the model is developed using the staggered grid finite volume method. The numerical scheme will be validated by comparing its results with the analytical solutions. As a result of the comparison, a rather excellent compatibility between the two results is achieved. Furthermore, the impacts that the friction coefficient has on the resonance phenomena are evaluated. It is observed that in the prevention of resonances, the bottom friction provides the best performance in the rectangular type while functioning the least efficient in the triangular basin. In addition, non-linearity effect as one of other factors that provide wave restriction is also considered and studied to compare its effect with the bottom friction effect on preventing resonance.     

矩形槽内水体大幅晃动分析中截断误差影响范围判断方法探讨

液体大幅晃动是目前CFD研究中一个热点。在CFD中,偏微分方程离散所产生的截断误差是主要的误差来源,且与网格尺度密切相关。本文首先从理论上介绍了截断误差的产生过程,由此过程可知截断误差与物理粘性的作用具有类似性。以此概念为基础,根据液体晃动动力响应的特点提出了在液体大幅晃动分析中,如何通过对物理粘性系数的参数敏感性分析,判断截断误差的作用范围从而判断计算精度是否满足要求的方法,最后与现有试验资料进行对比校核,验证了该方法的实用性。     

A second-order radiation boundary condition for the shallow water wave equations on two-dimensional unstructured finite element grids

A second-order radiation boundary condition (RBC) is derived for 2D shallow water problems posed in ‘wave equation’ form and is implemented within the Galerkin finite element framework. The RBC is derived by matching the dispersion relation for the interior wave equation with an approximate solution to the exterior problem for outgoing waves. The matching is correct to second order, accounting for curvature of the wave front and the geometry. Implementation is achieved by using the RBC as an evolution equation for the normal gradient on the boundary, coupled through the natural boundary integral of the Galerkin interior problem. The formulation is easily implemented on non-straight, unstructured meshes of simple elements. Test cases show fidelity to solutions obtained on extended meshes and improvement relative to simpler first-order RBCs.     

Accurate particle splitting for smoothed particle hydrodynamics in shallow water with shock capturing

The solution for the shallow water equations using smoothed particle hydrodynamics is attractive, being a mesh‐free, automatically adaptive method without special treatment for wet–dry interfaces. However, the relatively new method is limited by the variable kernel size or smoothing length being inversely proportional to water depth causing poor resolution at small depths. Boundary conditions at solid walls have also not been well resolved. To solve the resolution problem in small depths, a particle splitting procedure was developed (conveniently into seven particles), which conserves mass and momentum by varying the smoothing length, velocity and acceleration of each refined particle. This improves predictions in the shallowest depths where the error associated with splitting is reduced by one order of magnitude in comparison to other published works. To provide good shock capturing behaviour, particle interactions are treated as a Riemann problem with Monotone Upstream‐centred Scheme for Conservation Laws (MUSCL) reconstruction providing stability. For solid boundaries, the recent modified virtual boundary particle method was developed further to enable the zeroth moment to be accurately conserved where the smoothing length of particles is changing rapidly during particle splitting. The resulting method is applied to the one‐dimensional and the two‐dimensional axisymmetric wet‐bed dam break problems showing close agreement with analytical solutions, demonstrating the need for particle splitting. To demonstrate wetting and drying in a more complex case, the scheme is applied to oscillating water in a two‐dimensional parabolic basin and produces good agreement with the analytical solution. The method is finally applied to the European Concerted Action on DAm break Modelling dam‐break test case representative of realistic conditions and good predictions are made of experimental measurements with a 40% reduction in the computational time when particle splitting is employed. The overall method has thus become quite sophisticated but its generality and versatility will be attractive for various shallow water problems. Copyright © 2011 John Wiley & Sons, Ltd.     

2D numerical contributions for the study of non-cohesive sediment transport beneath tidal bores

2D numerical simulations of tidal bores were obtained using the OpenFOAM CFD software to solve the Navier–Stokes equations by means of the Finite Volume Method by applying a LES turbulence model. The trajectories of non-cohesive sediment particles beneath tidal bores were estimated using a tracker method. Using the fourth order Runge–Kutta scheme, the tracker method solves the Maxey and Riley equations, which requires the knowledge of the velocity field at time t. From 2D numerical simulations of tidal bores, we proposed a classification of tidal bores with respect to the Froude number Fr (or r the ratio of water depths). For a Froude number $1<Fr<1.43$ ($1<r<1.57$), the tidal bore is undular. For a Froude number $1.43<Fr<1.57$ ($1.57<r<1.75$), the tidal bore is partially breaking, which is similar to the transitional tidal bore defined by Furgerot (2014). And for a Froude number $Fr>1.57$ ($r>1.75$), the tidal bore is totally breaking. The numerical results of trajectories of non-cohesive sediment particles are similar to the type of trajectories given by the analytical model proposed by Chen et al. (2012) with some modifications to take into account the effects of gravity, elevation, and attenuation. The parameters of modified Chen's model, ${\beta }_1$, ${\beta }_2$ and ${\beta }_3$, are linearly proportional to the Froude number Fr. This is because the level of turbulence for undular tidal bores is low. The flow induced by an undular tidal bore is not complex. This physical phenomenon is quasi linear. The parameter ${\beta }_1$, related to the front celerity of the undular tidal bore, decreases when the Froude number Fr increases. The parameter ${\beta }_2$, related to the elevation, increases when the Froude number Fr increases. And the parameter ${\beta }_3$, related to the attenuation of the secondary waves, increases when the Froude number Fr increases.     

Adjoint Methods in Data Assimilation for Estimating Model Error

Data assimilation aims to incorporate measured observations into a dynamical system model in order to produce accurate estimates of all the current (and future) state variables of the system. The optimal estimates minimize a variational principle and can be found using adjoint methods. The model equations are treated as strong constraints on the problem. In reality, the model does not represent the system behaviour exactly and errors arise due to lack of resolution and inaccuracies in physical parameters, boundary conditions and forcing terms. A technique for estimating systematic and time-correlated errors as part of the variational assimilation procedure is described here. The modified method determines a correction term that compensates for model error and leads to improved predictions of the system states. The technique is illustrated in two test cases. Applications to the 1-D nonlinear shallow water equations demonstrate the effectiveness of the new procedure. This revised version was published online in July 2006 with corrections to the Cover Date.     

A method for analysing nesting techniques for the linearized shallow water equations

A method for analysing different nesting techniques for the linearized shallow water equations is presented. The problem is formulated as an eigenvector–eigenvalue problem. A necessary condition for stability is that the spectral radius of the propagation matrix is less than or equal to one. Two test cases are presented. The first test case is analysed, and effects of enforcing volume conservation and nudging in time are studied. A nesting technique is found that causes no growth of any eigenvectors for reasonable time steps. This nesting technique is then used on both test cases, and results are compared to an everywhere refined model and a coarse grid model. Copyright © 2002 John Wiley & Sons, Ltd.     

Numerical approximation of viscous terms in finite volume models for shallow waters

Two methods for the numerical treatment of viscous terms in shallow water equations are studied and computational details are given for structured grids. It is demonstrated that the first scheme, which is widely used, may lead to spurious oscillations arising from computational modes. In fact, the shortest resolvable waves of wave length 2Δx are invisible to this method. The second method, although more expensive, is free of computational modes and it presents a more accurate approximation of viscous terms. The dispersion relation of the second method is closer to the analytical case and it has a smaller truncation error, which is due to the fact that it uses a more localized control volume. Numerical experiments are also presented that support the study. Copyright © 2009 John Wiley & Sons, Ltd.     

Model of a strong discontinuity for the equations of spatial long waves propagating in a free-boundary shear flow

The long-wave equations describing three-dimensional shear wave motion of a free-surface ideal fluid are rearranged to a special form and used to describe discontinuous solutions. Relations at the discontinuity front are derived, and stability conditions for the discontinuity are formulated. The problem of determining the flow parameters behind the discontinuity front from known parameters before the front and specified velocity of motion of the front are investigated. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 49, No. 4, pp. 206–213, July–August, 2008.     

Emergent scale behaviour in the effective equations for an elastic metamaterial

Asymptotic homogenisation of an elastic metamaterial consisting of a series of plates interspaced by a fluid is considered. It is shown that the usual method must be extended by the inclusion of a second macroscale, the “emergent scale”, in order to correctly capture the behaviour of this metamaterial at low frequency. The leading order solutions for plane wave propagation are found and the effective constitutive equations derived. At leading order, the dispersion curves for plane wave propagation depend on the direction of propagation and the branch associated with the emergent scale is dispersive. The effective constitutive equations contain some terms, associated with the shear normal to the plates, for which stress is not proportional to strain but rather depends on the third derivative of the normal displacement with respect to transverse position. These terms are not present in the usual form of the Willis equations and the effective fields are non-analytic functions of frequency as the frequency tends to zero.