Irregular Triangular Grids Research Articles

Results of the Study of Resonant Oscillations in the Northern Part of the Shelf of the Peter the Great Gulf, the Sea of Japan

To study the features of resonant oscillations in the water column of the Peter the Great Gulf of the Sea of Japan, in situ measurements were carried out on its shelf, combined with numerical simulation of these processes. The observational data were obtained from autonomous bottom pressure gauges in Novik Bay in the winter of 2016. In the calculations, a spectral-difference model was used, modified to account for the ice cover, and implemented on an irregular triangular grid. The atmospheric forcing used in the model had periods from 15 to 55 min. As a result, characteristic series of spatio-temporal parameters for resonant oscillations of the studied water area were determined. The locations of the peaks on the simulated resonance curves correspond to the locations of well-defined maxima of the energy spectrum according to in situ measurements, hence indicating the possibility of a significant resonant amplification of level fluctuations by wave and periodic wind effects. The novelty of this study is inclusion of the winter period, when the surface of the bay is partially covered with ice.

Study of Free Oscillations of Bays in the Northwestern Part of Posyet Bay

To study the specific features of free surface oscillations in the northwestern part of Posyet Bay (the Sea of Japan), a series of experimental works using an installation with a laser meter for measuring hydrosphere pressure variations were carried out in 2012 and 2014. In the course of the joint analysis, measurement results for oscillations with periods of 10–30 min and the results of calculations using a numerical model of shallow water with a difference approximation on an irregular triangular space grid, datasets of the space–time parameters for the resonance oscillations of the studied water area were obtained. The results of the numerical simulations confirm the manifestation of the resonance properties of Novgorodskaya, Expedicii, and Reyd Pallada Bays water areas on the oscillations singled out during the experimental studies. The positions of the peaks on the model resonance curves are consistent with the positions of the clearly pronounced peaks of the energy spectrum in the field data.

A vertex-based reconstruction for cell-centered finite-volume discretization on unstructured grids

Recently, a vertex-based spatial reconstruction for unstructured cell-centered finite volume method has been proposed, showing advantages in accuracy, convergence, and efficiency. However, this method was only applied for inviscid flows. Moreover, for shock-capturing computations, the conventional cell-based limiter was the only choice to suppress numerical oscillations. In this work, the vertex-based reconstruction is extended for the solution of viscous flows, while automatically avoiding the typical decoupling problem. Moreover, a WENO-type nonlinear weighting strategy is designed and combined with the vertex-based reconstruction method, resulting in a simple and cost-efficient alternative to conventional slope limiters. In addition, the present method is combined with the recently proposed iterative near-boundary treatment, ensuring linear exactness even in the vicinity of boundaries, without sacrificing the overall computational efficiency. A series of numerical test cases involving viscous flows and shock waves provide evidence of the superior performance of the present method. The advantages of the present method are especially prominent on high aspect-ratio irregular triangular grids, which is a beneficial property for solving realistic fluid dynamic problems at high Reynolds numbers.

A hyperbolic Poisson solver for wall distance computation on irregular triangular grids

This paper presents an accurate and efficient Poisson solver for wall distance computations on irregular triangular grids. It is discussed that a typical formula of computing the wall distance from a numerical solution of a Poisson equation is not very accurate when solution contours are curved and the accuracy can be improved by a second-order formula derived by Taubin. However, the improved formula, as it requires second derivatives of the numerical solution, presents a challenge for a second-order method leading to zeroth-order accurate second-order derivatives on irregular grids. To overcome the difficulty, we develop a hyperbolic Poisson solver with gradients included as additional unknowns. In this method, the solution gradient can be obtained with second-order accuracy by a second-order discretization method, and therefore the second-derivatives can be obtained with first-order accuracy. Moreover, the hyperbolic method eliminates the numerical stiffness arising from the discretization of the Laplacian operator and achieves the ever-growing O(1/h) speed-up in iterative convergence, where h is a typical mesh spacing. Superior accuracy and efficiency of the developed method are demonstrated for irregular triangular grids.

Study of the Saffman–Taylor Instability in an Oil Reservoir Formation in Two Dimensions

The paper is devoted to the simulation of oil displacement by water and the formation of the Saffman–Taylor instability. The problem is solved in two-dimensional formulation. A circular domain with one injection well and eight production wells located along the contour around the injection well is considered as geometry. To study the patterns of oil displacement by water, hydrostatic pressure, oil and water seepage rates, and oil saturation are calculated. The graphical analysis considers mainly the oil saturation field. The hydrostatic pressure field is calculated by solving the steady-state piezoconductivity equation; the field of the oil-water seepage rate is calculated using the linear Darcy filtration law; and the oil saturation field is calculated from the solution of the advection transport equation. The two-phase nature of the flow considered in the problem lies in the fact that the oil and water phases have their own relative phase permeabilities calculated using the Brooks–Corey model. The equations are solved numerically using the finite volume method. An irregular triangular grid is used to discretize the computational domain. As a result of the simulation, it is established that the type of the Saffman–Taylor instability, due to its randomness, strongly depends on the computational grid. After the production wells are flooded, the displacement front stabilizes. The instability increases as the ratio of dynamic viscosities of oil and water increases.

Resonant oscillations in the western part of the Peter the Great Gulf in the Sea of Japan

The research aims to study the resonant properties of the water area of the Peter the Great Gulf, in the Sea of Japan. The coast of the gulf is exposed to the threat of tsunamis due to relatively high seismic activity in the Sea of Japan. On May 26, 1983, a tsunami hit several bays of the gulf with wave heights reaching 5 m. As a response, the coasts were equipped with three sophisticated sea level measuring stations. Field measurements with installation of autonomous pressure gauges were carried out August 18–26, 2016, to study the peculiarities of free surface oscillations in the western part of the Peter the Great Gulf. Spectral analysis of the recorded data and the data obtained from the Posyet automated post of the Russian Tsunami Warning Service showed, in particular, that there were several clearly pronounced peaks at periods of 40, 48, 79 and 89 min. These oscillations are coincident for the Sea of Japan and the studied water area with a wide entrance. A finite-volume numerical model was used to calculate free oscillations in the computational domain covering the Sea of Japan region. We performed model calculations with an irregular triangular grid. This allowed us to describe both the outer water area of the Sea of Japan and separate small-scale relief elements in the water area under study, which are important for the formation of local seiches. Isophases and spatial patterns of several resonant oscillations spectra at 15–105 min periods were obtained for the Peter the Great Gulf area. Model calculations carried out in this work showed an evident spatial structure of resonant oscillations. Such research, including field measurements in adjacent coastal waters, is required to identify and monitor potentially hazardous coastal areas. The amplification of resonant oscillations depends on how effectively the specific atmospheric disturbance generates corresponding eigenmodes of the Sea of Japan. In some bays, the amplitudes of the Sea of Japan eigen oscillations can increase due to resonant properties of coastal water areas, even with a low quality factor. If the amplitudes are large, such a double resonance can lead to a meteorological tsunami.

Global tide simulations with ICON-O: testing the model performance on highly irregular meshes

The global tide is simulated with the global ocean general circulation model ICON-O using a newly developed tidal module, which computes the full tidal potential. The simulated coastal M2 amplitudes, derived by a discrete Fourier transformation of the output sea level time series, are compared with the according values derived from satellite altimetry (TPXO-8 atlas). The experiments are repeated with four uniform and sixteen irregular triangular grids. The results show that the quality of the coastal tide simulation depends primarily on the coastal resolution and that the ocean interior can be resolved up to twenty times lower without causing considerable reductions in quality. The mesh transition zones between areas of different resolutions are formed by cell bisection and subsequent local spring optimisation tolerating a triangular cell’s maximum angle up to 84°. Numerical problems with these high-grade non-equiangular cells were not encountered. The results emphasise the numerical feasibility and potential efficiency of highly irregular computational meshes used by ICON-O.

On the loss and recovery of second-order accuracy with U-MUSCL

It is shown that U-MUSCL, an unstructured-grid extension of Van Leer's κ-scheme as proposed by Burg for the edge-based discretization [AIAA Paper 2005-4999], will not be exact for linear functions and thus will not be second-order accurate with a nonzero parameter χ if the flux quadrature point is not exactly halfway between two adjacent solution points as typical in a cell-centered finite-volume discretization on general unstructured grids, and moreover that it results, contrary to expectations, in less accurate solutions with χ=1/3 than with χ=0 on irregular triangular grids. To resolve the problem while keeping the flexibility of the original κ-scheme, we propose a simple technique to guarantee the linear exactness of U-MUSCL for any parameter value on arbitrary grids. Numerical results show that the improved scheme preserves second-order accuracy on irregular grids, successfully resolves the accuracy problem on irregular grids, and produces superior solutions with χ=1/3, as expected, even on unstructured grids.

Morphodynamic Responses to the Hong Kong–Zhuhai–Macao Bridge in the Pearl River Estuary, China

Li, M.; Li, W.; Xie, M., and Xu, T., 2021. Morphodynamic responses to the Hong Kong–Zhuhai–Macao Bridge in the Pearl River estuary, China. Journal of Coastal Research, 37(1), 168–178. Coconut Creek (Florida), ISSN 0749-0208.Human activities such as constructing large bridges, performing land reclamation, and creating harbors in complex tidal estuaries can result in severe and undesirable morphodynamic responses. This study investigates morphodynamic responses to the construction of the Hong Kong–Zhuhai–Macao Bridge (HZMB) in Lingdingyang Bay of the Pearl River estuary (LBPRE). The LBPRE in China is an important estuary with unique geomorphic features, complex hydrodynamic sediment conditions, dense shipping routes, high ship density, and a high navigation grade. The HZMB, a major sea-crossing transportation project consisting of bridges, a tunnel, and man-made islands, is being designed to cross the LBPRE. The morphodynamic responses resulting from the HZMB are likely to be significant. Therefore, it is necessary to carry out studies to optimize the HZMB design to minimize its impacts on hydrodynamics, sediment environment, harbors, and navigational channels as much as possible. In this article, the natural hydrodynamic and sediment conditions are first analyzed on the basis of in situ data. Next, a numerical model of tidal current, sediment movement, and seabed deformation based on TK-2D software is set up with an irregular triangular grid. The model is calibrated with in situ data, and the results show that the calculated tidal levels, tidal currents, suspended load concentrations, and seabed deformations are all in good agreement with field data. The model is then applied to carry out high-resolution simulations for design optimization of the HZMB from the perspective of its impacts on morphodynamics and hydrodynamics, as well as on harbors and channels. The study shows that the proposed HZMB design is feasible in terms of the morphodynamic responses. The conclusions provide an important foundation for the construction of the HZMB project.

A face-area-weighted ‘centroid’ formula for finite-volume method that improves skewness and convergence on triangular grids

This paper proposes a face-area-weighted ‘centroid’ as a superior alternative to the geometric centroid for defining a local origin in a cell-centered finite-volume method on triangular grids. It is demonstrated theoretically and numerically that the face-area-weighted ‘centroid’ can reduce grid skewness and improve iterative convergence for triangular grids. It is also shown that source terms do not have to be integrated over a cell and can be evaluated simply at the local origin without losing the design order of accuracy. Numerical results demonstrate that the face-area-weighted ‘centroid’ improves iterative convergence of an implicit defect-correction second-order finite-volume solver for inviscid and viscous flow problems on regular and irregular triangular grids.

An Entropy-Stable Residual Distribution Scheme for the System of Two-Dimensional Inviscid Shallow Water Equations

An entropy-stable residual distribution (RD) method is developed for the systems of two-dimensional shallow water equations (SWE). The construction of entropy stability for the residual distribution method is derived from finite volume method principles, albeit using a multidimensional approach. The paper delves into in-depth discussions on how finite volume methods achieve entropy stability in a “one-dimensional” sense for two-dimensional systems of SWE unlike the residual distribution methods. Results herein demonstrate the superiority of the entropy-stable RD methods relative to their finite volume counterparts, especially on highly irregular triangular grids. The comparative results with other established RD methods are also included, depicting similar performances with the Lax–Wendroff method for unsteady smooth flows but more accurate than the multidimensional upwind approaches (N, LDA) on both smooth and discontinuous test cases.

Studying Resonance Oscillations in the Eastern Part of the Posyet Bay

The field experiment with the installation of autonomous pressure gages in the Troitsa and Vityaz’ bays was held for studying the specific features of free surface oscillations in the Posyet Bay (the Sea of Japan). The interpretation and joint analysis of measurement data allowed obtaining the datasets of spatiotemporal parameters for free oscillations calculated using the spectral-difference model and for forced oscillations in the form of the response to the forcing with the periods of 10–50 minutes in the numerical model of shallow water with the difference approximation on the irregular triangular grid. The position of peaks on the model resonance curves and the values of frequencies of free oscillations are consistent with the position of clearly pronounced peaks of the energy spectrum from field data. This indicates the possibility of significant resonance enhancement of the amplitude of oscillations by periodic wind effects and incoming wave effects.

Consistent grid operators with the cell-nodal definition of grid functions

A principle of consistency for grid operators that ensures grid-operator inhomogeneous boundary-value problems are posed well is considered. Grid analogs of first-order differential operators and boundary operators that are consistent in the sense of fulfilling the grid analogs of integral relations are constructed on an irregular triangular grid. These relations are corollaries to the divergence theorem for vector fields that are the product of a scalar by a vector, the vector product of vectors, or the interior product of a vector by a dyadic. In each grid relation, one function is defined at the nodes; the other, in cells. Construction is performed using a grid-operator interpretation of the corollaries to the integral relations, which hold if one of the functions is a piecewise-linear interpolant to the nodal grid function, and the other is the piecewise-constant interpolant of the cell grid function.

A mass, energy, enstrophy and vorticity conserving (MEEVC) mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations

In this work we present a mimetic spectral element discretization for the 2D incompressible Navier–Stokes equations that in the limit of vanishing dissipation exactly preserves mass, kinetic energy, enstrophy and total vorticity on unstructured triangular grids. The essential ingredients to achieve this are: (i) a velocity–vorticity formulation in rotational form, (ii) a sequence of function spaces capable of exactly satisfying the divergence free nature of the velocity field, and (iii) a conserving time integrator. Proofs for the exact discrete conservation properties are presented together with numerical test cases on highly irregular triangular grids.

High-order shock-capturing hyperbolic residual-distribution schemes on irregular triangular grids

In this paper, we construct second- and third-order non-oscillatory shock-capturing hyperbolic residual-distribution schemes for irregular triangular grids, extending the schemes developed in J. Comput. Phys., 300 (2015), 455–491 to discontinuous problems. We present extended first-order N- and Rusanov-scheme formulations for a hyperbolic advection-diffusion system, and demonstrate that the hyperbolic diffusion term does not have any adverse effect on the solution of inviscid problems for a vanishingly small viscous coefficient. We then construct second- and third-order non-oscillatory hyperbolic residual-distribution schemes by blending the non-monotone second- and third-order schemes with the extended first-order schemes as typically done in the residual-distribution schemes, and examine them for discontinuous problems on irregular triangular grids. We also propose to use the Rusanov scheme to avoid non-physical shocks in combination with an improved characteristics-based nonlinear wave sensor for detecting shocks, compression, and expansion regions. We then verify the design order of accuracy of these blended schemes on irregular triangular grids.

Seiche oscillations in the Nakhodka Gulf

Spatiotemporal parameters of barotropic seiche oscillations in the Nakhodka Gulf in the Sea of Japan are considered using the spectral finite-difference model. The finite-difference approximation of shallow water equations is carried out at the irregular triangular spatial grid. The sets of periods and spatial structures of seiche oscillations corresponding to the strongly pronounced maxima of energy spectrum of sea level data from Nakhodka station of the Russian tsunami warning service are computed and presented.

Consistent grid analogs of invariant differential and boundary operators on an irregular triangular grid in the case of a grid nodal approximation

Grid analogs of invariant first-order differential operators and boundary operators are constructed on an irregular triangular grid. They are consistent in the sense of satisfying the grid analogs of integral relations that are consequences of the Gauss-Ostrogradskii formula for the divergence of a vector field as the product of a scalar and a vector, the vector product of vectors, or the inner product of a vector and a dyad. Construction is done using a grid-operator interpretation of integral relations written for piecewise linear interpolations of grid functions defined at the grid nodes. It is shown how to use consistent grid operators in constructing correct grid approximations of inhomogeneous boundary value problems at the operator level.

Numerical Study on Hydrodynamic Environment Effect of Channel-Narrowing Project in the Complex Tidal Estuary

The recent evolution of the North Branch of Changjiang Estuary is seriously affect the development of social economy in the region of Changjiang Estuary, regulation project of North Branch is extremely urgent. Now, the Channel-narrowing project in local implementation of North branch is already implemented. For study the effect of Channel-narrowing project on the hydrodynamic environment of the North Branch, a 2-D vertical average tidal current numerical model of Yangtze Estuary was set up with flexible irregular triangular grid to simulate the hydrodynamic before and after the Channel-narrowing Project in the North Branch. The following conclusions are drawn from the study. The results show that, the characteristics of tidal current in North Branches are not change a lot; the flood volume has been greatly decreased by the width of the North branch in the estuary area being narrowed down; the high tidal level lowers after the project. On the other hand, the low water level became increase and decrease, but the change of relative amplitude is not obvious. To sum up, the narrow project works to the advantage of the hydrodynamic environment in North branch.

First, second, and third order finite-volume schemes for advection–diffusion

In this paper, we present first, second, and third order implicit finite-volume solvers for advection–diffusion problems based on the first-order hyperbolic system method. In particular, we demonstrate that the construction of a uniformly accurate third-order advection–diffusion scheme is made trivial by the hyperbolic method while a naive construction of adding a third-order diffusion scheme to a third-order advection scheme can fail to yield third-order accuracy. We demonstrate also that the gradients are computed simultaneously to the same order of accuracy as that of the solution variable on irregular triangular grids: first, second and third order accurate gradients by the first, second, and third order schemes, respectively. Furthermore, the first and second order schemes are shown to achieve one order higher accuracy for the solution variable in the advection limit. It is also shown that these schemes are capable of producing highly accurate and smooth solution gradients along the boundary in a highly-skewed anisotropic irregular triangular grid while conventional schemes suffer from oscillations on such a grid. Numerical results show that these schemes are capable of delivering high accuracy over conventional schemes at a significantly reduced cost.

Numerical simulation of processes in solid deformable media in the presence of dynamic contacts using the grid-characteristic method

Several problems of dynamic deformation in the presence of movable contacting surfaces are considered. A grid-characteristic method is adapted using two-dimensional irregular triangular grids and a hybrid scheme of the first or second order of approximation. An algorithm for the calculation of sliding and static friction is proposed. We obtained the numerical solution for problems of wheel rolling, generation of shear waves, piston acceleration in a pipe, and ricochet.