Unstructured Triangular Grids Research Articles

Discretized pressure Poisson algorithm for steady incompressible flow on two-dimensional triangular unstructured grids

Abstract Many numerical algorithms have developed for Computational Fluid Dynamics (CFD). Previously, we provided a discretized pressure Poisson algorithm for steady incompressible flow on structured grids. Though this algorithm is very simple, there are still some problems for this algorithm to be implemented on unstructured grids. In order to solve incompressible flow problems on unstructured grids simply and easily, this paper developed some strategies to implement discretized pressure Poisson algorithm for steady incompressible flow on unstructured grids. Firstly, this paper discretized the convective term and diffusion term of momentum equations easily by the quantities of nodes and nearby points. Combined with the incenter-based Finite Volume Method (FVM), discretization of governing equations on triangular unstructured grids could be more easily implemented. Secondly, for the application of discretized pressure Poisson algorithm, this paper provided a discretization strategy for the continuity equation on unstructured grids. Thus, the mass conservation could be easily satisfied during the iteration of velocity calculation, which bring great advantages for the velocity calculation on unstructured grids. Then, discretized pressure Poisson algorithm was successfully used to calculate the lid-driven cavity flow problem on unstructured grids; the whole process was very simple and the results were also reliable.

Effects of complex boundary conditions on natural convection of a viscoplastic fluid

Purpose The purpose of this paper is to analyze the effects of complex boundary conditions on natural convection of a yield stress fluid in a square enclosure heated from below (uniformly and non-uniformly) and symmetrically cooled from the sides. Design/methodology/approach The governing equations are solved numerically subject to continuous and discontinuous Dirichlet boundary conditions by Galerkin’s weighted residuals scheme of finite element method and using a non-uniform unstructured triangular grid. Findings Results show that the overall heat transfer from the heated wall decreases in the case of non-uniform heating for both Newtonian and yield stress fluids. It is found that the effect of yield stress on heat transfer is almost similar in both uniform and non-uniform heating cases. The yield stress has a stabilizing effect, reducing the convection intensity in both cases. Above a certain value of yield number Y, heat transfer is only due to conduction. It is found that a transition of different modes of stability may occur as Rayleigh number changes; this fact gives rise to a discontinuity in the variation of critical yield number. Originality/value Besides the new numerical method based on the finite element and using a non-uniform unstructured grid for analyzing natural convection of viscoplastic materials with complex boundary conditions, the originality of the present work concerns the treatment of the yield stress fluids under the influence of complex boundary conditions.

A 2D vertical model for simulating surface and subsurface flows using finite element–finite volume methods

AbstractA numerical model is presented for simulation of hydrodynamics of a 2D vertical free surface domain consisting of an arbitrary partitioned porous and non-porous area. To this end, modified Navier–Stokes equations are considered which could be applied in surface water and in subsurface flows, simultaneously. A wide range of Reynolds number has been considered, from which non-Darcy effects have also been taken into account. A fractional step method has been used in the time discretization procedure, where the convection and diffusion terms are separated from the pressure term while solving the momentum equations. To include the variation of surface elevation in computation, the domain has been divided into two parts, namely, ‘interior subdomain’, which never gets dry during the simulation period, covered by fixed unstructured triangular grids and ‘top layer’ with only a one layer structured grid, the position of which varies with the water surface. The validation of the proposed model has been achieved by comparison of its results with both theoretical and experimental data reported in the literature.

Моделювання літньої циркуляції та розподілу температури і солоності влітку в морі Беллінсгаузена та на шельфі Антарктичного півострова

The objective of the study is to simulate the summer circulation and the distribution of temperature and salinity in the Bellingshausen Sea and on the shelf of the Antarctic Peninsula. Numerical methods are used for simulation by means the SCHISM model with an unstructured triangular horizontal grid and a vertical local sigma coordinate system (LSC2). The k-kl model of turbulence was used to describe turbulence. Heat, momentum and salt fluxes were set on the ocean surface, which were calculated according to the ERA-Interim reanalysis. At open boundaries, the vertical distribution of temperature and salinity was determined according to the HYCOM reanalysis calculations. On the western open border, the level deviations and the vertical velocity distribution calculated by HYCOM were also specified. At the open eastern boundary of the computational domain, the radiation conditions were specified. The results of the HYCOM reanalysis were also used as initial conditions. The results of modelling the current, temperature, and salinity fields in the period February—March 2014 were compared with the available observational data on the shelf of the Antarctic Peninsula, including data from the Ukrainian Antarctic expedition. The calculated vertical distributions of temperature and salinity on the shelf are consistent with observations, in particular, in the position and magnitude of the minimum temperature of the cold intermediate layer and its temperature in the region of the Argentine Islands. Calculations have shown that the main component of the circulation is the Antarctic Circumpolar Current (ACC), which carries water to the east. In ACC, several jets are formed flowing into the Drake Strait, whereas a chain of vortices separates the shelf zone of the Antarctic Peninsula and the ACC. In the summer, a current directed southwards along the edge of the shelf, caused mainly by large-scale ocean circulation and the wind field. It was concluded that the formation of zones of upwelling and downwelling on the shelf of the Antarctic Peninsula substantially depends on the changes in the atmospheric circulation over the Bellingshausen Sea.

Simulation of multi-component multi-phase fluid flow in two-dimensional anisotropic heterogeneous porous media using high-order control volume distributed methods

In this paper, flow of multi-component two-phase fluids in highly heterogeneous anisotropic two-dimensional porous media is studied using computational methods suitable for unstructured triangular and/or quadrilateral grids. The physical model accounts for miscibility and compressibility of fluids while gravity and capillary effects are neglected. The governing equations consist of a pressure equation together with a system of mass conservation equations. For solving pressure equation, a new method called Control Volume Distributed Finite Element Method (CVDFEM) is introduced which uses Control Volume Distributed (CVD) vertex-centered grids. It is shown that the proposed method is able to approximate the pressure field in highly anisotropic and heterogeneous porous media fairly accurately. Moreover the system of mass conservation equations is solved using various upwind and central schemes. These schemes are extended from one-dimensional to two-dimensional unstructured grids. Using a series of numerical test cases, comparison is made between different approaches for approximation of the hyperbolic flux function. Semi one-dimensional high-order data reconstruction procedures are employed to decrease stream-wise numerical diffusion. The results suggest that the Modified Dominant Wave (MDW) scheme outperforms other hyperbolic schemes studied in this paper from both accuracy and computational cost points of view.

Laminar natural convection of yield stress fluids in annular spaces between concentric cylinders

This paper aims to investigate the laminar natural convection of Bingham fluids between horizontal concentric cylinders. The governing equations are solved numerically by using the Galerkin’s weighted residuals scheme of finite element method with a non-uniform unstructured triangular grid. The effects of various parameters such as the Rayleigh number (103≤Ra≤106), yield number (0≤Y≤Ymax), ratio of gap width to inner cylinder radius (0.5≤L/Ri≤5) and fixed Prandtl number (Pr=100 since results are not sensitive to Pr in the tested range) on the flow and temperature profiles as well as on the heat transfer and morphology of yielded/unyielded regions are presented and discussed. The heat transfer rate shows a positive dependence on the Rayleigh number and L/Ri, however, there is an optimum value for L/Ri which maximizes the heat transfer rate. The results also show that the yield number has a significant impact on the flow and temperature fields and the heat transfer performance. In fact, the heat transfer rate decreases with the rise of the yield number and above a certain value of Y, the convection does not occur, and heat transfer is only due to conduction. It is found that while the enhancement of the Rayleigh number augments the heat transfer, it is not necessarily associated with a decrease in the size of the unyielded zones.

A goal-oriented and self-adaptive mesh refinement approach for the even parity neutron transport equation

This survey focuses on a goal-oriented approach for the adaptive mesh refinement (AMR) of even parity neutron transport equation on the unstructured triangular or quadrilateral grids. Two a posteriori error estimates are compared for conducting the refinement process. User specified quantity of interest is set as the source in the adjoint equation to obtain a map of importance for steering the AMR procedure toward the desired computational target. Both forward and adjoint even parity equations are solved and the discontinuity of the flux gradient across the border of an element is used for the evaluation of element's error. The self-adaptivity is provided through a fully automated cycle in which the h-refinement is performed until the error in the quantity of interest drops below a user prescribed threshold. Several numerical test cases are investigated to assess the performance of the proposed approach in both fixed source and eigenvalue search problems against uniform mesh refinement.

An accurate and robust AUSM-family scheme on two-dimensional triangular grids

Most numerical schemes for solving high-speed compressible flow problems exhibit an instability that usually occurs inside the numerical shock structure in low-dissipative shock-capturing finite volume methods. In examining several test cases, the flux-difference splitting and the AUSM family of schemes cannot satisfy the robustness requirement, which manifests as the carbuncle phenomenon on two-dimensional triangular grids. This paper presents an accurate and robust AUSM-family scheme ( $$\hbox {AUSMDV}^+$$ scheme) that is verified against shock-induced anomalies on two-dimensional triangular grids. The linearized discrete analysis of an odd–even decoupling problem is applied to investigate the perturbation damping mechanism of these schemes. The corresponding recursive equations show that the $$\hbox {AUSMDV}^+$$ scheme is less sensitive to these anomalies than are other schemes in the AUSM family. Finally, the presented scheme is extended to achieve second-order solution accuracy. Its robustness and efficiency are then evaluated on both structured and unstructured triangular grids. The $$\hbox {AUSMDV}^+$$ scheme yields a physically meaningful solution for all test cases without introducing an additional shock fix technique.

Flood Modeling in Urban Area Using a Well-Balanced Discontinuous Galerkin Scheme on Unstructured Triangular Grids

Flood Modeling in Urban Area Using a Well-Balanced Discontinuous Galerkin Scheme on Unstructured Triangular Grids

Impact of spatially varying hydraulic conductivities on tumor interstitial fluid pressure distribution

Abstract This work deals with numerical computation of the interstitial fluid pressure distribution for various synthetic tumor configurations in two spatial dimensions. In order to attend a reasonable degree of realism, we also consider asymmetric configurations and use unstructured triangular grids to accurately discretize the computational domains. A multi point flux approximation method, previously used for porous medium flow simulations, is utilized for solving the corresponding Poisson equation for the pressure. We are particularly interested in the effects of spatial variation of the hydraulic conductivity as well as vessel hydraulic conductivities on the interstitial fluid pressure. Spatially varying hydraulic conductivity was first studied by Liu and Schlesinger [17] in some special situations using analytical methods. Here we complement their work by performing a set of relatively comprehensive numerical investigations. A considerable difference in the computed pressure profiles is found when comparing the results with the more classical approach using piecewise constant hydraulic conductivities. We also explore the response of the numerical model when varying i.e. size of necrotic core and vessel hydraulic conductivities. The pressure profiles display a striking dependence on the arterial hydraulic conductivity in the peripheral region. A strong dependence on the arterial hydraulic conductivity is also observed when introducing so called “vascular hot-spots” in the model as well as looking at non-symmetric geometries. Finally, we discuss our numerical results in light of other numerical studies of TIFP distribution and experimental studies of tumor biology. It is concluded that the computed results are in accordance with the general understanding of TIFP. Although we have worked on a simplified system, our results highlight the importance of describing hydraulic conductivity as a spatially varying parameter.

A 3D parallel boundary element method on unstructured triangular grids for fully nonlinear wave-body interactions

This paper presents the development and validation of a three-dimensional numerical wave tank devoted to studying wave-structure interaction problems. It is based on the fully nonlinear potential flow theory, here solved by a boundary element approach and using unstructured triangular meshes of the domain's boundaries. Time updating is based on a second-order explicit Taylor series expansion. The method is parallelized using the Message Passing Interface (MPI) in order to take advantage of multi-processor systems. For radiation problems, with cylindrical bodies moving in prescribed motion, the free-surface is updated with a fully Lagrangian scheme, and is able to reproduce reference results for nonlinear forces exerted on the moving body. For diffraction problems, semi-Lagrangian time-updating is used, and reproduces nonlinear effects for diffraction on monopiles. Finally, we study the nonlinear wave loads on a fixed semi-submersible structure, thereby illustrating the possibility to apply the proposed numerical model for the design of offshore structures and floaters.

An Assessment of Two Models of Wave Propagation in an Estuary Protected by Breakwaters

Breakwaters influence coastal wave climate and circulation by blocking and dissipating wave energy. In a large harbor, these effects are combined with wave generation, refraction and reflection. Accurate representation of these processes is essential to the determination of coastal circulation and wave processes. MIKE21SW and SWAN are two third-generation spectral wave models which are used widely in coastal research and engineering applications. Recently improved versions of the models are able to consider the influence of breakwater structures. In this study, we used available observations to evaluate the accuracy of model simulations of waves in New Haven Harbor, Connecticut, USA, an estuary with three detached breakwaters near the mouth. The models were executed on their optimum unstructured triangular grid. The boundary conditions were derived from a bottom mounted Acoustic Doppler Current Profilers (ADCP) on the offshore side of the breakwaters. Wind forcing was applied using data from the Central Long Island Sound buoy. We found that both models were largely consistent with observations during storms. However, MIKE21SW predicted some of storm peaks slightly better. SWAN required the finer grid to achieve the optimum condition, but as it uses a fast, fully implicit algorithm, the computational times were similar. Also, the sensitivity analysis represents that wind forcing and the breakwaters have significant impact on the results.

Validation of a two-dimensional gas-kinetic scheme for compressible natural convection on structured and unstructured meshes

Gas-kinetic schemes (GKS) have been developed as a kinetic Finite-Volume approach to computational fluid dynamics. The GKS a priori allows to obtain approximate solutions of the fully compressible Navier-Stokes equations. In our contribution we show simulation results of compressible natural convection at large temperature differences and low Mach numbers beyond the applicable range of the Boussinesq approximation. The simulations were performed on non-uniform quadrilateral and unstructured triangular grids. The dependence of the critical Rayleigh number on the temperature difference for compressible Rayleigh-Bénard convection, predicted by theory, is accurately reproduced. Moreover, heat transfer in a buoyancy driven square cavity with differentially heated sides at large Rayleigh numbers and large temperature differences is investigated. Temperature and velocity profiles as well as Nusselt numbers show good agreement with benchmark results in literature. After validating the scheme for thermal compressible convection, we investigate unsteady natural convection at a Rayleigh number of Ra=5⋅109 and at a large temperature difference of Th/Tc=4. We find that compressibility has a leading influence on the stability of the boundary layers, such that the flow at the heated wall becomes unstable, whereas the flow at the cooled wall remains stable. This phenomenon has not yet received much attention.

Numerical investigation of the effects of aquatic plants on wind-induced currents in Taihu Lake in China

Shallow lakes are an important part of the earth's water circulation system, and many factors may impact the evolution of the lake water circulation, including the wind, the topography, the human activities, and the aquatic plants. This paper proposes a depth-averaged 2-D hydrodynamic model to investigate the interaction of the wind-induced current and the aquatic plant in the lake. The model is based on the generalized shallow water equations solved by an explicit finite volume method with unstructured triangular grids. The drag force of the vegetation is considered into the momentum equations as the source term. Remote sensing techniques are applied to evaluate the aquatic vegetation in the Taihu Lake, China, based on Landsat TM satellite images. The study model is then used to simulate the characteristics of the wind-induced currents in the Taihu Lake, without and with the vegetation effects. The simulation results are in good agreement with the field measurements, demonstrating that the aquatic plants significantly affect the magnitude of the velocity and the flow circulation induced by the wind in the Taihu Lake. In addition, a sensitivity analysis reveals that the plant parameters (the density and the drag force coefficient) are significant factors influencing the velocity and the structure of the currents in the Taihu Lake.

Bernstein–Bézier based finite elements for efficient solution of short wave problems

In this work, the Bernstein–Bézier Finite Element Method (BBFEM) is implemented to solve short wave problems governed by the Helmholtz equation on unstructured triangular mesh grids. As for the hierarchical Finite Element (FE) approach, this high order FE method benefits from the use of static condensation which is an efficient tool for reducing the total number of degrees of freedom and bandwidth of high order FE global matrices. The performance of BBFEM with static condensation (BBFEMs) is assessed via three benchmark problems and compared to that of the Partition of Unity Finite Element Method (PUFEM) in terms of accuracy, conditioning and memory requirement. Numerical results dealing with problems of Hankel sources interference and wave scattering by a rigid cylinder on quasi-uniform mesh grids indicate that BBFEMs is able to achieve a better accuracy but PUFEM is slightly better conditioned when the wave is not well resolved. However, with a sufficient wave resolution, BBFEMs is better conditioned than PUFEM. Results from L-shaped domain problem, with non quasi-uniform mesh grids, show that the conditioning of BBFEMs remains reasonable while PUFEM with large numbers of enriching plane waves on mesh grids locally well resolved leads to ill-conditioning.

Well-balanced positivity preserving central-upwind scheme with a novel wet/dry reconstruction on triangular grids for the Saint-Venant system

In this paper, we construct an improved well-balanced positivity preserving central-upwind scheme for the two-dimensional Saint-Venant system of shallow water equations. As in Bryson et al. (2011) [7], our scheme is based on a continuous piecewise linear discretization of the bottom topography over an unstructured triangular grid. The main new technique is a special reconstruction of the water surface in partially flooded cells. This reconstruction is an extension of the one-dimensional wet/dry reconstruction from Bollermann et al. (2013) [3]. The positivity of the computed water depth is enforced using the “draining” time-step technique introduced in Bollermann et al. (2011) [4]. The performance of the proposed central-upwind scheme is tested on a number of numerical experiments.

Numerical modeling of wave-current interaction in Merak Port, Indonesia

This study aimed to investigate the effects of currents to wave height in the waters of Merak Port, Banten using wave spectrum model. The wave-current interaction simulation was performed by utilizing SWAN (Simulating Waves Nearshore) in stationary mode and using the unstructured triangular grid. Data of wind velocity and direction as model inputs were obtained from BMKG (Indonesian Agency for Meteorology, Climatology, and Geophysics), whereas tidal current velocity data was obtained from TMD (Tide Model Driver). The model was simulated in two different scenarios, i.e. without and with following (adverse) current which is same (opposite) with wave propagation. The model input of dominant west wind was used to represent the west monsoon, while the dominant wind from the north was used to represent the east monsoon. The simulation results show that the maximum wave height is 2.98 m in the west monsoon and 1.9 m in east monsoon. In the west (east) monsoon the following current decrease wave height to 9.26% (5.09%), while the adverse current increase wave height up to 9.07% (6.56%).

Induced polarization in a 2.5D marine controlled-source electromagnetic field based on the adaptive finite-element method

The induced polarization (IP) in rocks and minerals is of significance to the marine controlled-source electromagnetic (CSEM) field. We propose an adaptive finite-element algorithm for the 2.5D frequency-domain forward modeling of marine CSEM that considers the induced polarization. The geoelectrical model is discretized using an unstructured triangular elemental grid that accommodates the complex topography and geoelectrical structures. We use the Cole–Cole model to describe the IP and develop a complex resistivity forward modeling algorithm. We compare the simulation results with published 1D model results and subsequently calculate the electromagnetic field for variable azimuth sources, IP parameters, and topography. Finally, we analyze the IP effect on the marine CSEM field and show that IP of oil reservoirs and topography affects the marine CSEM electromagnetic field.

A comparative study of boundary conditions for the density-based solvers in the framework of OpenFOAM

In the framework of OpenFOAM, two methods for the boundary condition treatments are discussed and compared for the density-based solvers, including the strong- and the weak-imposition approach. For the weak-imposition approach, the Riemann method is implemented. For the treatment of subsonic inlet and outlet in the Riemann method, the nonreflective boundary condition based on the finite wave model is adopted and extended to the unstructured grid. With the Riemann method, the boundary condition treatments are transformed into the acquisition of the numerical flux with approximate Riemann solvers at the boundary face, same as that of interior faces. And the information of propagating waves at the boundary is correctly reflected through the usage of approximate Riemann solvers. What is more attractive is that the formal accuracy near the boundary as well as the global accuracy can be preserved without destroying the numerical stability of the simulation. For the strong-imposition approach, the ghost cell method is implemented. In spite of its simplicity, the robustness is not good enough on the unstructured triangular grid. Besides, it has been found that the formal accuracy near the boundary is only first order, thus influencing the solution quality and the global accuracy of the flowfields. Some typical inviscid examples are presented to evaluate the performance of these two methods.

Numerical Modeling of Plasma Devices by the Particle-In-Cell Method on Unstructured Grids

The paper considers methods and algorithms providing the basis for a computer program implementing an axial-symmetric electrostatic version of the particle-in-cell method on unstructured triangular grids. In the presented implementation, the Poisson equation is approximated using the finite volume method. A discrete analog of the Poisson equation is solved by the multigrid method. Charged particle trajectories are calculated using the Boris method. Methods for interpolating electrostatic fields on unstructured grids and obtaining the charge density in the computational domain are considered. Special attention is paid to the specifics of implementing these methods in axisymmetric geometry. The developed computer code is tested on the problem of a flat diode operating in the space charge mode.