Unstructured Mesh Finite Volume Research Articles

Inverse distance interpolation for used in unstructured mesh finite volume solver

This article discusses adjusting inverse distance interpolation for use in unstructured mesh finite volume solutions. The adjustment was made on the weight function of the inverse distance interpolation using the Laplacian of the flow variable inside a Voronoi-dual of finite volume cells. We tested the accuracy of the adjusted inverse distance interpolation on two-dimensional potential flows. It was found that the adjusted and standard inverse distance interpolations have a similar degree of accuracy when used in unstructured, Delaunay based, finite volume mesh. However, the L1 norm error of the adjusted version of the inverse distance interpolation was much smaller than the L1 norm error of the standard version.

Accuracy of Cell Centres to Vertices Interpolation for Unstructured Mesh Finite Volume Solver

Accuracy of Cell Centres to Vertices Interpolation for Unstructured Mesh Finite Volume Solver

Accuracy of compact-stencil interpolation algorithms for unstructured mesh finite volume solver

This study considers the accuracy of cell-to-face centre interpolation of convected quantities in unstructured finite volume meshes with cell-centred storage of variables. The accuracy of the interpolation algorithms were tested in isolation using ideal data to determine their numerical accuracy on both standard and artificially distorted meshes. It was found that the formally second- and third-order accurate interpolations based on one-dimensional interpolation along the line connecting the cells to the right and left of the face under consideration only have first-order accuracy in standard unstructured mesh, and less than first-order accuracy in distorted unstructured mesh. L1 interpolation errors in the distorted unstructured mesh are greater than in standard unstructured mesh. The order of accuracy and L1 errors can be improved by applying spatial corrections. The formally second-order accurate multi-dimensional interpolations tested in this study that are not based on one-dimensional interpolation along lines connecting the neighbour cells have first-order accuracy in both standard and distorted unstructured mesh. Linear interpolation between end vertices produces greatest L1 error in standard mesh; polynomial interpolation, linear interpolation between cell centres and standard QUICK produce the greatest L1 error in distorted mesh. Spatially correct QUICK, spatially correct linear interpolation between cell centres, Laplacian interpolation to face centres, and Taylor series expansion about an upstream cell produce the smallest L1 error in both standard and distorted mesh. Based on accuracy and the simplicity of implementation, Taylor series expansion about an upstream cell is the best choice for use in unstructured mesh.

Unstructured mesh finite volume methods for fractional-in-space nonlinear reaction–diffusion equations on irregular domains

Fractional models have been used as a powerful tool to study anomalous diffusion in human brains or heart issues with structural heterogeneity. However, the biological issues are always with irregular shapes. Many published methods for solving the fractional models are no longer effective for irregular domains. In this paper, using a finite volume method based on an unstructured mesh, we numerically solve a space fractional nonlinear reaction–diffusion model based on a fractional Laplacian operator (−∇2)α∕2 and nonlinear source term. The highlight of this method is that the considered solution domains can be arbitrarily shaped due to the fact that the meshes are unstructured, which is important in real applications. Here, we consider two dimensions fractional-in-space nonlinear reaction–diffusion equations (FISNRDE-2D) with homogeneous Neumann boundary conditions. The efficiency of our method is showcased by solving the fractional-in-space Allen–Cahn model, the Fractional FitzHugh–Nagumo model and the fractional Gray–Scott model.

A finite volume procedure for fluid flow, heat transfer and solid-body stress analysis

ABSTRACTA unified cell-centered unstructured mesh finite volume procedure is presented for fluid flow, heat transfer and solid-body stress analysis. An in-house procedure (A. W. Date, Solution of Transport Equations on Unstructured Meshes with Cell-Centered Colocated Variables. Part I: Discretization, International Journal of Heat and Mass Transfer, vol. 48 (6), 1117-1127, 2005) is extended to include the solid-body stress analysis. The transport terms for a cell-face are evaluated in a structured grid-like manner. The Cartesian gradients at the center of each cell-face are evaluated using the coordinate transformation relations. The accuracy of the procedure is demonstrated by solving several benchmark problems involving different boundary conditions, source terms, and types of loading.

Adjoint-Based Functional Correction for Unstructured Mesh Finite Volume Methods

The goal of this paper is to demonstrate the use of adjoint-based functional correction and mesh adaptation for aerodynamic flows with an unstructured mesh finite volume solver. A key feature of our approach is that all calculations are performed on a single mesh, unlike other error correction and mesh adaptation schemes. As using the original p-truncation error estimate is not helpful in improving the functional, we use a smoothed estimate of the truncation error to correct the functional for both inviscid and viscous flows. The correction term is based on the smoothed truncation error and the adjoint solution, with both the continuous and discrete adjoints. The same correction term is used as an adaptation indicator for goal-based mesh adaptation as well. Our results show the effectiveness of our method in improving the convergence rate for test cases of interest in computational aerodynamics.

Mesh-Resolved Airfoil Simulations Using Finite Volume and Discontinuous Galerkin Solvers

A traditional second-order accurate finite volume unstructured mesh solver and a high-order discontinuous Galerkin solver are used to simulate turbulent flow over a NACA 0012 airfoil. Using a family of standardized high-density grids, mesh-resolved solutions are obtained. The flow is simulated by solving the Reynolds-averaged Navier–Stokes equations closed by the negative Spalart–Allmaras turbulence model. The flow conditions for this case are , , and . Lift, drag, pitching moment, pressure, and skin friction coefficients are provided for multiple grids and discretization orders and are compared against other simulation results from well known solvers. The current simulations give very similar results to these benchmark solvers, pointing toward fully mesh-resolved simulations and providing verification evidence of correct and consistent implementation of these discretizations. Results obtained using the high-order discontinuous Galerkin discretizations show higher accuracy using fewer degrees of freedom compared to the finite volume discretization and the benchmark results. Also, it is shown for the discontinuous Galerkin simulations that refinement converges quicker to the mesh-resolved solutions compared to refinement.

An effective 3D leapfrog scheme for electromagnetic modelling of arbitrary shaped dielectric objects using unstructured meshes

In computational electromagnetics, the advantages of the standard Yee algorithm are its simplicity and its low computational costs. However, because of the accuracy losses resulting from the staircased representation of curved interfaces, it is normally not the method of choice for modelling electromagnetic interactions with objects of arbitrary shape. For these problems, an unstructured mesh finite volume time domain method is often employed, although the scheme does not satisfy the divergence free condition at the discrete level. In this paper, we generalize the standard Yee algorithm for use on unstructured meshes and solve the problem concerning the loss of accuracy linked to staircasing, while preserving the divergence free nature of the algorithm. The scheme is implemented on high quality primal Delaunay and dual Voronoi meshes. The performance of the approach was validated in previous work by simulating the scattering of electromagnetic waves by spherical 3D PEC objects in free space. In this paper we demonstrate the performance of this scheme for penetration problems in lossy dielectrics using a new averaging technique for Delaunay and Voronoi edges at the interface. A detailed explanation of the implementation of the method, and a demonstration of the quality of the results obtained for transmittance and scattering simulations by 3D objects of arbitrary shapes, are presented.

2D Unstructured Mesh Finite Volume Method for Simulating Structural Dynamics

A novel procedure for calculating the dynamic response of elastic solid structures is presented. The ultimate aim of this study is to develop a consistent set of finite volume (FV) methods on unstructured meshes for the analysis of dynamic fluidstructure interaction (FSI). This paper describes a two-dimensional (2D) FV cell-vertex based method for dynamic solid mechanics. A novel matrix-free implicit scheme was developed using the Newmark method and dual time step algorithm and the model is validated with a 2D cantilever test case as well as a 2D plate one.

Influence of sea surface wind wave turbulence upon wind-induced circulation, tide–surge interaction and bed stress

A three-dimensional finite volume unstructured mesh model of the west coast of Britain, with high resolution in the coastal regions, is used to investigate the role of wind wave turbulence and wind and tide forced currents in producing maximum bed stress in the eastern Irish Sea. The spatial distribution of the maximum bed stress, which is important in sediment transport problems, is determined, together with how it is modified by the direction of wind forced currents, tide–surge interaction and a surface source of wind wave turbulence associated with wave breaking. Initial calculations show that to first order the distribution of maximum bed stress is determined by the tide. However, since maximum sediment transport occurs at times of episodic events, such as storm surges, their effects upon maximum bed stresses are examined for the case of strong northerly, southerly and westerly wind forcing. Calculations show that due to tide–surge interaction both the tidal distribution and the surge are modified by non-linear effects. Consequently, the magnitude and spatial distribution of maximum bed stress during major wind events depends upon wind direction. In addition calculations show that a surface source of turbulence due to wind wave breaking in shallow water can influence the maximum bed stress. In turn, this influences the wind forced flow and hence the movement of suspended sediment. Calculations of the spatial variability of maximum bed stress indicate the level of measurements required for model validation.

Modeling of 2D density-dependent flow and transport in porous media using finite volume method

A two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular mesh is developed for modeling density-dependent flow and transport through saturated–unsaturated porous media. The combined flow and transport model can handle a wide range of real-word problems, including the simulation of flow alone, contaminant transport alone, and combined flow and transport. Saltwater intrusion problems and instability caused by denser water on the top were investigated in this paper. Because the fundamental mechanism causing saltwater intrusion most likely is caused by density-induced convection and dispersion, the developed model is used to assess the interplay between density-driven flow and subsurface media through which the saltwater intrusion occurs. The mathematical formulation of the model is comprised of fluid flow and solute transport equations, coupled by fluid density. In the specific case of saltwater intrusion and unstable brine transport problems, this set of governing equation is non-linear and requires iterative methods to solve them simultaneously. Three case studies, which include a wide range of physical conditions, are used for verification of presented model and comparison with previously published solutions from other researchers presents encouraging agreements.

Computational modelling of variably saturated flow in porous media with complex three‐dimensional geometries

AbstractA computational procedure is presented for solving complex variably saturated flows in porous media, that may easily be implemented into existing conventional finite‐volume‐based computational fluid dynamics codes, so that their functionality might be geared upon to readily enable the modelling of a complex suite of interacting fluid, thermal and chemical reaction process physics. This procedure has been integrated within a multi‐physics finite volume unstructured mesh framework, allowing arbitrarily complex three‐dimensional geometries to be modelled. The model is particularly targeted at ore heap‐leaching processes, which encounter complex flow problems, such as infiltration into dry soil, drainage, perched water tables and flow through heterogeneous materials, but is equally applicable to any process involving flow through porous media, such as in environmental recovery processes. The computational procedure is based on the mixed form of the classical Richards equation, employing an adaptive transformed mixed algorithm that is numerically robust and significantly reduces compute (or CPU) time. The computational procedure is accurate (compares well with other methods and analytical data), comprehensive (representing any kind of porous flow model), and is computationally efficient. As such, this procedure provides a suitable basis for the implementation of large‐scale industrial heap‐leach models. Copyright © 2005 John Wiley & Sons, Ltd.

Evaluation of adaptive mesh refinement and coarsening for the computation of compressible flows on unstructured meshes

AbstractThe compressible gas flows of interest to aerospace applications often involve situations where shock and expansion waves are present. Decreasing the characteristic dimension of the computational cells in the vicinity of shock waves improves the quality of the computed flows. This reduction in size may be accomplished by the use of mesh adaption procedures. In this paper an analysis is presented of an adaptive mesh scheme developed for an unstructured mesh finite volume upwind computer code. This scheme is tailored to refine or coarsen the computational mesh where gradients of the flow properties are respectively high or low. The refinement and coarsening procedures are applied to the classical gas dynamic problems of the stabilization of shock waves by solid bodies. In particular, situations where oblique shock waves interact with an expansion fan and where bow shocks arise around solid bodies are considered. The effectiveness of the scheme in reducing the computational time, while increasing the solution accuracy, is assessed. It is shown that the refinement procedure alone leads to a number of computational cells which is 20% larger than when alternate passes of refinement and coarsening are used. Accordingly, a reduction of computational time of the same order of magnitude is obtained. Copyright © 2005 John Wiley & Sons, Ltd.

A finite volume unstructured mesh approach to dynamic fluid–structure interaction: an assessment of the challenge of predicting the onset of flutter

Computational modelling of dynamic fluid–structure interaction (DFSI) is a considerable challenge. Our approach to this class of problems involves the use of a single software framework for all the phenomena involved, employing finite volume methods on unstructured meshes in three dimensions. This method enables time and space accurate calculations in a consistent manner. One key application of DFSI simulation is the analysis of the onset of flutter in aircraft wings, where the work of Yates et al. [Measured and Calculated Subsonic and Transonic Flutter Characteristics of a 45° degree Sweptback Wing Planform in Air and Freon-12 in the Langley Transonic Dynamic Tunnel. NASA Technical Note D-1616, 1963] on the AGARD 445.6 wing planform still provides the most comprehensive benchmark data available. This paper presents the results of a significant effort to model the onset of flutter for the AGARD 445.6 wing planform geometry. A series of key issues needs to be addressed for this computational approach. • The advantage of using a single mesh, in order to eliminate numerical problems when applying boundary conditions at the fluid-structure interface, is counteracted by the challenge of generating a suitably high quality mesh in both the fluid and structural domains. • The computational effort for this DFSI procedure, in terms of run time and memory requirements, is very significant. Practical simulations require even finer meshes and shorter time steps, requiring parallel implementation for operation on large, high performance parallel systems. • The consistency and completeness of the AGARD data in the public domain is inadequate for use in the validation of DFSI codes when predicting the onset of flutter.

An unstructured mesh finite volume method for modelling saltwater intrusion into coastal aquifers

In this paper, a two-dimensional finite volume unstructured mesh method (FVUM) based on a triangular background interpolation mesh is developed for analysing the evolution of the saltwater intrusion into single and multiple coastal aquifer systems. The model formulation consists of a ground-water flow equation and a salt transport equation. These coupled and non-linear partial differential equations are transformed by FVUM into a system of differential/algebraic equations, which is solved using backward differentiation formulas of order one through five. Simulation results are compared with previously published solutions where good agreement is observed.

Dynamic fluid–structure interaction using finite volume unstructured mesh procedures

A three-dimensional finite volume, unstructured mesh (FV-UM) method for dynamic fluid–structure interaction (DFSI) is described. Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. It involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge. Until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. More recently, strategies for solving the full coupling between the fluid and solid mechanics behaviour have been developed. A key contribution has been made by Farhat et al. [Int. J. Numer. Meth. Fluids 21 (1995) 807] employing FV-UM methods for solving the Euler flow equations and a conventional finite element method for the elastic solid mechanics and the spring based mesh procedure of Batina [AIAA paper 0115, 1989] for mesh movement. In this paper, we describe an approach which broadly exploits the three field strategy described by Farhat for fluid flow, structural dynamics and mesh movement but, in the context of DFSI, contains a number of novel features: • a single mesh covering the entire domain, • a Navier–Stokes flow, • a single FV-UM discretisation approach for both the flow and solid mechanics procedures, • an implicit predictor–corrector version of the Newmark algorithm, • a single code embedding the whole strategy.

Details of an integrated approach to three- dimensional dynamic fluid structure interaction

Fluid structure interaction, as applied to flexible structures, has wide application in diverse areas such as flutter in aircraft, wind response of buildings, flows in elastic pipes and blood vessels. Numerical modelling of dynamic fluid-structure interaction (DFSI) involves the coupling of fluid flow and structural mechanics, two fields that are conventionally modelled using two dissimilar methods, thus a single comprehensive computational model of both phenomena is a considerable challenge and until recently work in this area focused on one phenomenon and represented the behaviour of the other more simply. A single, finite volume unstructured mesh (FV-UM) spatial discretisation method has been employed on a single mesh for the entire domain. The Navier Stokes equations for fluid flow are solved using a SIMPLE type procedure and the Newmark b algorithm is employed for solving the dynamic equilibrium equations for linear elastic solid mechanics and mesh movement is achieved using a spring based mesh procedure for dynamic mesh movement. In the paper we describe a number of additional computation issues for the efficient and accurate modelling of three-dimensional, dynamic fluid-structure interaction problems.

Numerical modelling and validation of Marangoni and surface tension phenomena using the finite volume method

Surface tension induced flow is implemented into a numerical modelling framework and validated for a number of test cases. Finite volume unstructured mesh techniques are used to discretize the mass, momentum and energy conservation equations in three dimensions. An explicit approach is used to include the effect of surface tension forces on the flow profile and final shape of a liquid domain. Validation of this approach is made against both analytical and experimental data. Finally, the method is used to model the wetting balance test for solder alloy material, where model predictions are used to gain a greater insight into this process. Copyright © 2000 John Wiley & Sons, Ltd.

Multiphysics modelling of the metals casting process

Metals casting is a process governed by the interaction of a range of physical phenomena. Most computational models of this process address only what are conventionally regarded as the primary phenomena-heat conduction and solidification. However, to predict the formation of porosity (a factor of key importance in cast quality) requires the modelling of the interaction of the fluid flow, heat transfer, solidification and the development of stress-deformation in the solidified part of a component. In this paper, a model of the casting process is described which addresses all the main continuum phenomena involved in a coupled manner. The model is solved numerically using novel finite volume unstructured mesh techniques, and then applied to both the prediction of shape deformation (plus the subsequent formation of a gap at the metal-mould interface and its impact on the heat transfer behaviour) and porosity formation in solidifying metal components. Although the porosity prediction model is phenomenologically simplistic it is based on the interaction of the continuum phenomena and yields good comparisons with available experimental results. This work represents the first of the next generation of casting simulation tools to predict aspects of the structure of cast components.

Solving convection and convection-reaction problems using the method of lines

The application of software based on the method of lines (MOL) to convection and convection-reaction problems is considered. The components of a prototype software package, SPRINT2D, are unstructured mesh finite volume algorithms, coupled spatial and temporal error control algorithms, and modified time integration techniques for the large systems of ordinary differential equations. The software is applied to a realistic combustion problem and the lessons learned used to discuss the validity of the MOL approach.