Unstructured Finite Volume Method Research Articles

Numerical investigation of the scour around a diamond- and square-shaped pile in a narrow channel

In this paper, a numerical investigation is conducted to study the scour dynamics around vertical square- and diamond-shaped piles in clear-water and live-bed conditions. The focus is particularly on cases involving narrow channels, aiming to examine the differences compared to more extensively studied scenarios, such as wide channels with circular piles. This work employs a Large Eddy Simulation (LES) model to simulate the turbulent flows around the structures. The hydrodynamic model is based on the Navier–Stokes equations in σ-coordinate, and the bed evolution is simulated by solving the Exner-Polya equation. The governing equations are solved using a second-order unstructured finite-volume method. The model is initially validated through a comparison with experimental data and numerical results found in the literature. Subsequently, it is applied to conduct a comprehensive study, enhancing our understanding of the various conditions influencing the scour process. The results indicate that the shape of the piles has different effects on the scour depth under clear-water and live-bed conditions. In live-bed conditions, the simulations reveal a deeper scour around the square-shaped pile compared to the diamond-shaped pile at the equilibrium stage. Conversely, an opposite behavior is observed in clear-water conditions, with a deeper scour around the diamond pile. Moreover, this behavior is consistent across both narrow and wide channels. It is discovered that, in clear-water conditions, the evolution of the scour in a narrow channel reaches equilibrium faster than in a wide channel, leading to a deeper equilibrium scour hole in the wide channel scenario.

Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods

Corrected Linear-Galerkin Schemes to Preserve Second-Order Accuracy for Cell-Centered Unstructured Finite Volume Methods

3D Heat Conduction Solver for Complex Geometries

Abstract: The essential pre-requisites for any solver to tackle real life 3D problems are to handle complex geometries and quick turnaround time. In this direction an Unstructured Finite Volume (FVM) based multi-threaded heat conduction solver has been developed to address both the issues respectively. The solver is object oriented and written in Java language. Unstructured Finite Volume method has been utilized for the space discretization and a four stage Runge-Kutta scheme is used for time integration. This solution methodology is highly parallelizable and extensively used in CFD codes. The present solver can handle non-linear steady state and transient 3D thermal problems with Dirichlet, Neumann and surface radiation boundary conditions. The solver has been validated against bench mark problems. In the present work a 3D complex geometry is discretised into 2.5 lakh hexahedrons has been taken as a test case to demonstrate the efficiency of the solver with multi-threading in dual processor machines. The results obtained are presented.

Analytical Solution of Vertically Integrated Equations of a Confined Aquifer in two-dimension using Finite Volume method

Groundwater plays an important role in feeding many families and societies whose access to water is a huge problem for several reasons, then the mathematical model is an essential part of a hydrologist’s daily life to understand the physical phenomena. In this paper, we used Dupuit’s assumption to reduce the groundwater flow equation and integrated vertically it to obtainthe diffusivity equation in terms of the hydraulic head, further, the introduction of fundamental ideas in groundwater research, including vocabulary and mathematics will give readers the basis knowledge. Numerous numerical techniques can be recognized, the unstructured finite volume method was used to solve a transient diffusivity equation through a coastal confined aquifer. Theconstructed finite volume schemes are used to study the distribution of hydraulic heads in the model domain, to determine water table fluctuations, and to predict the drawdown phenomena.

Numerical Study of the Flow and Blockage Ratio of Cylindrical Pier Local Scour

A three-dimensional large eddy simulation model is used to simulate the turbulent flow dynamics around a circular pier in live-bed and clear-water scour conditions. The Navier–Stokes equations are transformed into a σ-coordinate system and solved using a second-order unstructured triangular finite-volume method. We simulate the bed evolution by solving the Exner-Polya equation assisted by a sand-slide model as a correction method. The bedload transport rate is based on the model of Engelund and Fredsœ. The model was validated for live-bed conditions in a wide channel and clear-water conditions in a narrow channel against the experimental data found in the literature. The in-house model NSMP3D can successfully produce both the live-bed and clear-water scouring throughout a stable long-term simulation. The flow model was used to study the effects of the blockage ratio in the flow near the pier in clear-water conditions, particularly the contraction effect at the zone where the scour hole starts to form. The scour depth in the clear water simulations is generally deeper than the live-bed simulations. In clear-water, the results show that the present model is able to qualitatively and quantitatively capture the hydrodynamic and morphodynamic processes near the bed. In comparison to the wide channel situation, the simulations indicate that the scour rate is faster in the narrow channel case.

An unstructured finite-volume level set / front tracking method for two-phase flows with large density-ratios

We extend the unstructured LEvel set / froNT tracking (LENT) method [1,2] for handling two-phase flows with strongly different densities (high-density ratios) by providing the theoretical basis for the numerical consistency between the mass and momentum conservation in the collocated Finite Volume discretization of the single-field two-phase Navier-Stokes equations. Our analysis provides the theoretical basis for the mass conservation equation introduced by Ghods and Herrmann [3] and used in [4–8]. We use a mass flux that is consistent with mass conservation in the implicit Finite Volume discretization of the two-phase momentum convection term, and solve the single-field Navier-Stokes equations with our SAAMPLE segregated solution algorithm [2]. The proposed ρLENT method recovers exact numerical stability for the two-phase momentum advection of a spherical droplet with density ratios ρ−/ρ+∈[1,104]. Numerical stability is demonstrated for in terms of the relative L∞ velocity error norm, for density-ratios in the range of [1,104], dynamic viscosity-ratios in the range of [1,104] and very strong surface tension forces, for challenging mercury/air and water/air fluid pairings. In addition, the solver performs well in cases characterized by strong interaction between two phases, i.e., oscillating droplets and rising bubbles. The proposed ρLENT method1 is applicable to any other two-phase flow simulation method that discretizes the single-field two-phase Navier-Stokes Equations using the collocated unstructured Finite Volume Method but does not solve an advection equation for the phase indicator using a flux-based approach, by adding the proposed geometrical approximation of the mass flux and the auxiliary mass conservation equation to the solution algorithm.

Numerical Modeling of Supersonic Combustion of Hydrogen in the Wake Region of a Strut

Numerical modeling of combustion of a sonic hydrogen stream injected into a supersonic stream of air using a strut is presented. Results are obtained by numerically solving unsteady, compressible, turbulent Navier-Stokes equations, using Unstructured Finite Volume Method (UFVM) incorporating RNG based κ−ε two equation model and time integration using three stage Runge Kutta method. Reaction is modeled using an eight step H2-air chemistry. The developed numerical procedure is based on the implicit treatment of chemical source terms by preconditioning and solved along with unsteady turbulent Navier Stokes equations explicitly. Code is validated against a standard wall jet experimental data and is successfully used to model the reacting flow field resulting due to combustion of hydrogen in the wake region of strut placed in heated supersonic air stream.

Cost vs Accuracy: DNS of turbulent flow over a sphere using structured immersed-boundary, unstructured finite-volume, and spectral-element methods

We report a comparative study of three numerical solvers for the direct numerical simulation of the flow over a sphere at Re = 3700. A high-order spectral-element code (Nek5000), a general purpose, unstructured finite-volume solver (OpenFOAM) and an in-house Cartesian solver using the immersed-boundary method (IBM) are employed for the analysis; results are compared against previous numerical and experimental data. Numerical results show that Nek5000 and the IBM code operate within a similar computational performance range, in terms of cost-vs-accuracy analysis, for both global parameters as well as local flow features. On the other hand, OpenFOAM needed a significantly higher number of degrees of freedom (and, overall, a higher cost) to match some of the basic features of the flow, such as the length of the recirculation bubble forming downstream the sphere. For the finest grid resolutions, the three codes are in good agreement for most of the analyzed flow metrics. Overall, our results suggest that high-order methods and second-order, energy-conserving approaches based on the IBM may be both viable options for high-fidelity scale-resolving simulations of turbulent flows with separation.

A parallel unstructured multi-color SOR solver for 3D Navier–Stokes equations on graphics processing units

In this paper, we develop a GPU parallelization technique to accelerate the pressure Poisson solver. We apply it in three-dimensional Navier–Stokes simulations with arbitrary geometries. The numerical method is based on a projection method, and the discretization is obtained by combining a second-order unstructured finite-volume method and a σ transformation. The pressure solution is responsible for this formulation’s most time-consuming computation tasks. This is because the Poisson equation must be approximated. This work proposes a GPU-based Multi-Color Successive Over-Relaxation (MC-SOR) method to solve the resulting sparse linear system. The proposed parallel finite-volume code is verified and validated by solving the Poisson problem. We employ analytical solutions and well-known benchmark problems with complex flows. These include the Taylor–Green vortex problem and the problem of turbulent flow around a circular cylinder.

A constrained boundary gradient reconstruction method for unstructured finite volume discretization of the Euler equations

We propose a novel boundary gradient and solution reconstruction approach for the second-order cell-centered unstructured finite volume method for solving the inviscid Euler equations. The proposed constrained reconstruction method involving ghost cells is shown to be more accurate than several conventional boundary reconstruction methods and linear-exact in the near-boundary areas. In the present procedure, the solutions at the ghost cell centroids are given by solving constraint equations, which are established based on boundary conditions. At the same time, the solution gradients in boundary cells and the solutions in boundary surfaces are obtained altogether, while guaranteeing the compatibility of the boundary conditions with the reconstructed results. A variety of numerical test cases show that the new method effectively reduces the errors of near-boundary approximations and improves the overall numerical accuracy. Furthermore, because the constrained reconstruction is only used for the boundary cells, the extra computational cost barely changes the overall computational efficiency.

Numerical assessment of the potential stream energy extraction of a tidal lagoon inlet

A hypothetical hydrodynamic basin that imitated the Nador lagoon with its inlet was developed to estimate the potential stream energy of the Nador lagoon during a one-month hydrodynamic simulation. The water level rise in the numerical basin was investigated using a depth-averaged hydrodynamic model. The model is based on an unstructured finite volume method to solve the hyperbolic shallow water equations. The method uses an efficient Riemann solver that is balanced with bottom elevation to approximate the convective term. A second Runge-Kutta technique is used to guarantee time marching. Numerical experiments were made to calibrate the model by controlling the bottom frictions in order to simulate the hydrodynamic behaviors. The numerical results of tidal elevation are compared with those from field observations. Good agreements between the numerical model and measurements data were founded. Corresponding current velocity is then used to evaluate potential energy extraction. The results show that the candidate site holds a power density that can achieve 1.246 KW/m2.

Estimation of traffic emissions in a polycentric urban city based on a macroscopic approach

This paper presents a macroscopic dynamic modeling framework for road traffic emissions in an urban city that has numerous central business districts (CBDs). The framework combines a two-dimensional (2D) continuum dynamic traffic assignment (DTA) model that captures macroscopic features of traffic flow and a microscopic instantaneous emission model associated with the instantaneous speed and acceleration of traffic flow. A heterogeneous traveling population continuously distributed in space is classified according to the trip destinations chosen by travelers. Each copy of flow tends to select an optimal route with the minimum cost to destination by means of instantaneous traffic information. For each copy of flow, an unstructured finite volume method as well as a fast sweeping method is designed to solve the proposed model. In order to validate the feasibility of the model and algorithm, a case study is conducted as an example to study the macroscopic characteristics of traffic flow and predict traffic-related pollutants like NOx, VOC, CO2 and PM in urban areas.

Numerical modeling of two dimensional non-capacity model for sediment transport by an unstructured finite volume method with a new discretization of the source term

The main goal of this work is the resolution of the two-dimensional shallow water equations of water–sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and Runge–Kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.

Computational analysis of cross‐flow of power‐law fluids through a periodic square array of circular cylinders

AbstractThe steady flow of non‐Newtonian power‐law fluids across a periodic square array of infinitely long circular cylinders is studied numerically using an unstructured finite volume method. The local and global flow characteristics have extensively been explored by the systematic variations of the pertinent dimensionless parameters as follows: fluid volume fraction (ϕf = .70–.99), Reynolds number (Re = 1–40), and power‐law index (n = .4–1.8). Qualitatively, the dense and curved streamlines are seen with the increasing inertial forces and shear‐thinning behavior across all the fluid volume fractions. Further, the pressure coefficient over the surface of periodic cylinders is significantly influenced by the governing parameters and found to be maximum and minimum for the upstream and downstream cylinders, respectively. The individual and total drag coefficients have shown complex dependence on n, ϕf, and Re. For shear‐thinning fluids (n < 1), the pressure drag coefficient dominates over the friction drag coefficient, whereas an opposite response is seen for the shear‐thickening fluids (n > 1) except at Re = 40. Further, both individual and total drag coefficients are observed to increase and decrease in the ranges of ϕf of .70–.90 and .92–.99, respectively, with increasing n. The strong interactions between the periodic cylinders, at smaller ϕf, diminish with a corresponding increase in ϕf. In addition, simple predictive correlations for the pressure, friction, and total drag coefficients have been developed to gain further physical insights into the detailed flow kinematics. Finally, the present findings display a good agreement with the available literature.

Three-Dimensional Hydrostatic Curved Channel Flow Simulations Using Non-Staggered Triangular Grids

Non-staggered triangular grids have many advantages in performing river or ocean modeling with the finite-volume method. However, horizontal divergence errors may occur, especially in large-scale hydrostatic calculations with centrifugal acceleration. This paper proposes an unstructured finite-volume method with a filtered scheme to mitigate the divergence noise and avoid further influencing the velocities and water elevation. In hydrostatic pressure calculations, we apply the proposed method to three-dimensional curved channel flows. Approximations reduce the numerical errors after filtering the horizontal divergence operator, and the approximation is second-order accurate. Numerical results for the channel flow accurately calculate the velocity profile and surface elevation at different Froude numbers. Moreover, secondary flow features such as the vortex pattern and its movement along the channel sections are also well captured.

Numerical Study and Economy Analysis of Two Heated Crude Oil Pipelines Laid in One Ditch

In this paper, the transportation economy of two heated crude oil pipelines laid in one ditch is analyzed by taking into account the influence of operating temperature, interval between two pipelines, and distance between two heating stations on the heating energy consumption. To analyze the transportation economy, the two heated crude oil pipelines laid in one ditch are simulated under four operating conditions based on an unstructured finite volume method. Compared with laying two crude oil pipelines separately in two ditches, the results attest notably higher soil temperature, meaning reduced heat dissipation of each pipeline by laying two pipelines in one ditch. It is inferred that for the same desired oil temperature at the inlet of the next heating station, laying two pipelines in one ditch requires lower oil temperature at the outlet of heating station, indicating decreased energy cost at the heating station and improved transportation economy. Then economy analysis of four configurations of laying two pipelines in one ditch is performed. By comparing the results of four conditions, the interval between two pipelines of 1.2 m is found to save the energy most efficiently, which is as large as 26.6% compared with that of laying two pipelines in two separate ditches. In addition, narrowing the pipeline interval and extending the distance between heating stations is beneficial to save heating energy. This study is expected to provide valuable guidance for operation optimization of heated crude oil pipelines.

Unstructured Finite-Volume Model of Sediment Scouring Due to Wave Impact on Vertical Seawalls

The numerical modeling of sediment transport under wave impact is challenging because of the complex nature of the triple wave–structure–sediment interaction. This study presents three-dimensional numerical modeling of sediment scouring due to non-breaking wave impact on a vertical seawall. The Navier–Stokes–Exner equations are approximated to calculate the full evolution of flow fields and morphodynamic responses. The bed erosion model is based on the van Rijn formulation with a mass-conservative sand-slide algorithm. The numerical solution is obtained by using a projection method and a fully implicit second-order unstructured finite-volume method in a σ-coordinate computational domain. This coordinate system is employed to accurately represent the free-surface elevation and sediment/water interface evolution. Experimental results of the velocity field, surface wave motion, and scour hole formation hole are used to compare and demonstrate the proposed numerical method’s capabilities to model the seawall scour.

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.

Effects of mesh loop modes on performance of unstructured finite volume GPU simulations

In unstructured finite volume method, loop on different mesh components such as cells, faces, nodes, etc is used widely for the traversal of data. Mesh loop results in direct or indirect data access that affects data locality significantly. By loop on mesh, many threads accessing the same data lead to data dependence. Both data locality and data dependence play an important part in the performance of GPU simulations. For optimizing a GPU-accelerated unstructured finite volume Computational Fluid Dynamics (CFD) program, the performance of hot spots under different loops on cells, faces, and nodes is evaluated on Nvidia Tesla V100 and K80. Numerical tests under different mesh scales show that the effects of mesh loop modes are different on data locality and data dependence. Specifically, face loop makes the best data locality, so long as access to face data exists in kernels. Cell loop brings the smallest overheads due to non-coalescing data access, when both cell and node data are used in computing without face data. Cell loop owns the best performance in the condition that only indirect access of cell data exists in kernels. Atomic operations reduced the performance of kernels largely in K80, which is not obvious on V100. With the suitable mesh loop mode in all kernels, the overall performance of GPU simulations can be increased by 15%-20%. Finally, the program on a single GPU V100 can achieve maximum 21.7 and average 14.1 speed up compared with 28 MPI tasks on two Intel CPUs Xeon Gold 6132.

A conservative level-set/finite-volume method on unstructured grids based on a central interpolation scheme

In this paper, we introduce a second-order unstructured finite-volume method developed to solve a conservative level-set equation in two- and three-dimensional geometries. The characteristic function for the flow is defined at the center of each cell while the face-normal velocities are calculated at the mid-points of the corresponding cell faces. An interpolation method independent of cell shape and based on a central scheme is applied for the approximations at the cell faces. The capabilities and performance of the proposed scheme are validated using several 2D and 3D tests. Results show that the present numerical method is quite suitable for working within the smooth framework resulting from the signed distance function. Moreover, this method yields very accurate results in interface-capturing problems such as the single vortex deformation.