Control Volume Faces Research Articles

Application of a Novel Weighted Essentially Non-Oscillatory Scheme for Reynolds-Averaged Stress Model and Reynolds-Averaged Stress Model/Large Eddy Simulation (RANS/LES) Coupled Simulations in Turbomachinery Flows

In numerical simulations, achieving high accuracy without significantly increasing computational cost is often a challenge. To address this issue, this paper proposes an improved finite volume Weighted Essentially Non-Oscillatory (WENO) scheme for structured grids. By employing a single-point quadrature rule to perform flux integration on the control volume faces, this scheme is designed for use in NUAA-Turbo three-dimensional fluid solvers based on structured grids, utilizing RANS and RANS/LES coupling to simulate turbomachinery flows. Firstly, the new WENO scheme is validated against classical numerical test cases to evaluate its stability and reliability in handling discontinuities, double Mach reflection problems, and Rayleigh–Taylor (RT) instability. Compared to the original scheme, this improved finite-volume WENO scheme demonstrates better stability near discontinuities and more effectively resolves flow features at the same grid resolution. Next, for engineering applications related to turbomachinery, such as compressor and turbine characteristics, calculations using RANS are performed and the results obtained with WENO-ZQ3 and WENO-JS3 are compared. Finally, the new fifth-order WENO scheme is applied to RANS/LES coupling simulations of turbine wake and film cooling. The results indicate that the improved finite-volume WENO scheme provides better stability and accuracy in engineering applications. For instance, the average error in calculating compressor efficiency characteristics is reduced from 0.76% to 0.05%, the error in turbine vane pressure distribution compared to the experimental values is within 1%, and the error in film cooling efficiency centerline distribution compared to the experimental values is within 3%. Additionally, the qualitative results of turbine wake and film cooling show that even with a small number of grid points, more detailed flow physics can be captured, thereby reducing computational costs in aerodynamic applications.

Second-order, positive, and unconditional energy dissipative scheme for modified Poisson–Nernst–Planck equations

First-order energy dissipative schemes in time are available in literature for the Poisson–Nernst–Planck (PNP) equations, but second-order ones are still in lack. This work proposes novel second-order discretization in time and finite volume discretization in space for modified PNP equations that incorporate effects arising from ionic steric interactions and dielectric inhomogeneity. A multislope method on unstructured meshes is proposed to reconstruct positive, accurate approximations of mobilities on faces of control volumes. Numerical analysis proves that the proposed numerical schemes are able to unconditionally ensure the existence of positive numerical solutions, original energy dissipation, mass conservation, and preservation of steady states at discrete level. Extensive numerical simulations are conducted to demonstrate numerical accuracy and performance in preserving properties of physical significance. Applications in ion permeation through a 3D nanopore show that the modified PNP model, equipped with the proposed schemes, has promising applications in the investigation of ion selectivity and rectification. The proposed second-order discretization can be extended to design temporal second-order schemes with original energy dissipation for a type of gradient flow problems with entropy.

Mixed Convection in a Square Cavity Filled with Porous Medium with Bottom Wall Periodic Boundary Condition

Transient mixed convection heat transfer in a confined porous medium heated at periodic sinusoidal heat flux is investigated numerically in the present paper. The Poisson-type pressure equation, resulted from the substituting of the momentum Darcy equation in the continuity equation, was discretized by using finite volume technique. The energy equation was solved by a fully implicit control volume-based finite difference formulation for the diffusion terms with the use of the quadratic upstream interpolation for convective kinetics scheme to discretize the convective terms and the temperature values at the control volume faces. The numerical study covers a range of the hydrostatic pressure sinusoidal amplitude range and time period values of . Numerical results show that the pressure contours lines are influenced by hydrostatic head variation and not affected with the sinusoidal amplitude and time period variation. It is found that the average Nusselt number decreases with time and pressure head increasing and decreases periodically with time and amplitude increasing. The time averaged Nusselt number decreases with imposed sinusoidal amplitude and cycle time period increasing.

Evaluation of Interpolation Scheme Alternatives and Variation of the Number of Slave Cells in a Cut-Cell Methodology

The present work determines numerical solutions applied to flow problems in a cut-cell framework, introducing and evaluating two interpolation alternatives for the treatment of the convective terms and the effect of the variation of the number of slave cells generated near the solid interfaces. Using the upwind, QUICK and WAHYD (TVD) schemes, three benchmark cases were studied in the laminar regime, namely, flow between concentric cylindrical walls, flow in an inclined channel and flow around a cylinder. The numerical results obtained were favorable for the proposed interpolation methodology that prevents velocity over/under-estimations on the finite control volume faces, observing a tendency to produce smaller errors and mid-to-high computational efficiencies when coupled with a smaller number of slave cells generated at the boundaries. Although the magnitude of the errors found were small, improvements are of more significance for quantities that depend on gradient estimations at surfaces.

On the boundary conditions of magnetic field in OpenFOAM and a magnetic field solver for multi-region applications

This work begins with the discretization of the governing equations of magnetic field for different type of magnetic media over a two-dimensional rectangular domain using a Cartesian grid system. The basic rules that the coefficients of the discretization equation should obey, to ensure physical realism and overall balance, are discussed. In order to satisfy the rule, “consistency at control-volume faces”, the formulation of the permeability at the interface between two adjacent control volumes is derived. It reveals that the interface permeability is equal to the anti-linear interpolation of that on the cell centroids under the assumption of flux consistency at control-volume faces. For the purpose of effectively calculating magnetic fields using the FVM based OpenFOAM toolbox, the representations of boundary conditions on the interfaces between different type of media are for the first time derived. A C++ class named magPhiMixedFvPatchField describing the newly derived boundary conditions is defined based on OpenFOAM libraries. The new open-source solver labeled as magneticMultiRegionFoam, for the simulation of static magnetic field with multi-regions, based on FVM in the OpenFOAM framework is for the first time developed and validated. Comparison of the results of the predicted magnetic field for a cylindrical magnet in air with those obtained from Comsol software and the analytical method verifies the fidelity of the solver implementation. For the case of calculating the magnetic levitation forces exerted on a magnet immersed in ferrofluid, the results obtained from magneticMultiRegionFoam show better consistency with the experimental ones than Comsol software. Program summaryProgram Title: magneticMultiRegionFoamCPC Library link to program files:https://doi.org/10.17632/858w9rpf58.1Licensing provisions: GNU General Public License v3Programming language: C++External routines/libraries: OpenFOAM (http://www.openfoam.org)Nature of problem: Solving the magnetic field with multi-regionsSolution method: Finite Volume Method (FVM) with a self-defined class describing the boundary conditions between different type of magnetic media.Additional comments including restrictions and unusual features: The current version of the solver can only be applied at static magnetic field.

Three-dimensional unstructured gridding for complex wells and geological features in subsurface reservoirs, with CVD-MPFA discretization performance

Grid generation for reservoir simulation, must honour classical key geological features and multilateral wells. The features to be honoured are classified into two groups; (1) involving layers, faults, pinchouts and fractures, and (2) involving well distributions. In the former, control-volume boundary aligned grids (BAGs) are required, while in the latter, control-point (defined as the centroid of the control-volume) well aligned grids (WAGs) are required. Depending on discretization method type and formulation, a choice of control-point and control-volume type is made, i.e. for a cell-centered method the primal grid cells act as control-volumes, otherwise for a vertex-centered method the dual-grid cells act as control-volumes. Novel three-dimensional unstructured grid generation methods are proposed that automate control-volume boundary alignment to geological features and control point alignment to complex wells, yielding essentially perpendicular bisector (PEBI) meshes either with respect to primal or dual-cells depending on grid type. Both grid types use tetrahedra, pyramids, prisms and hexahedra as grid elements. Primal-cell feature aligned grids are generated using special boundary surface protection techniques together with constrained cell-centered well trajectory alignment. Dual-cell feature aligned grids are generated from underlying primal-meshes, whereby features are protected such that dual-cell control-volume faces are aligned with interior feature boundaries, together with protected vertex-centered (control point) well trajectory alignment. The novel methods of grid generation presented enable practical application of both method types in 3-D for the first time. The primal and dual grids generated here demonstrate the gridding methods, and enable the first comparative performance study of cell-vertex versus cell-centered control-volume distributed multi-point flux approximation (CVD-MPFA) finite-volume formulations using equivalent mesh resolution on challenging problems in 3-D. Pressure fields computed by the cell-centered and vertex-centered CVD-MPFA schemes are compared and contrasted relative to the respective degrees of freedom employed, and demonstrate the relative benefits of each approximation type. Stability limits of the methods are also explored. For a given mesh the cell-vertex method uses approximately a fifth of the unknowns used by a cell-centered method and proves to be the most beneficial with respect to accuracy and efficiency. Numerical results show that vertex-centered CVD-MPFA methods outperform cell-centered CVD-MPFA method.

Generation of Water Waves Using Momentum Source Wave‐Maker Applied to a RANS Solver

Nowadays, as the development of Computational Fluid Dynamics (CFD) and the numerical wave tank (NWT) has advanced, numerical analysis has become increasingly useful and powerful for the ship designing and ship hydrodynamics. In this study, a momentum source wave‐maker and an analytical relaxation wave absorber were embedded into 2D RANS equation model with RSM turbulence closure scheme to establish the NWT for ship designing and hydrodynamics. The VOF (volume‐of‐fluid) method was applied to accurately capture the water free surface. The body force‐weighted scheme is chosen for pressure interpolation and the second order upwind scheme for discretization of the momentum equation. In order to calculate convection and diffusion fluxes through the control volume faces, PISO algorithm is adopted for pressure‐velocity coupling. The momentum source function for wave generation and the analytical relaxation function for wave absorption were deduced for constructing the NWT (numerical wave tank). The proposed NWT was then validated by the laboratory measurements of Umeyama and the analytical solution, indicating that the constructed NWT is effective and accurate.

Testing of various schemes with quasi-one-dimensional reconstruction of gasdynamic variables in the case of unstructured-grid calculations

Several schemes of the second-order approximation worked out in the literature for unstructured-grid-based computations of gasdynamic flows are described. The convective fluxes on the control-volume's faces are evaluated using Roe's approximate Riemann solver. The MUSCL approach with the use of various quasi-one-dimensional schemes of reconstruction of gasdynamic variables and limiters making the solution monotonic is applied in order to improve the approximation accuracy. Comparative analysis of the working capacity of the schemes under consideration has been carried out through solving two problems of inviscid gas flow. Namely, the transonic NACA-0012 airfoil flow and the superpersonic flow in a duct with a central ramp were computed. The smoothness of solution, obtained with different schemes, dissipativity features of the schemes and computational process stability were evaluated.

A simplified enhanced MPFA formulation for the elliptic equation on general grids

Multi-point flux approximation (MPFA) has proven to be a powerful tool for discretizing the diffusion equation on general grids with heterogeneous anisotropic permeability tensors and hence, removing the O(1) error introduced by two-point flux approximation (TPFA) for non-K-orthogonal grids. However, it is well known that the classical MPFA-O suffers from monotonicity issues and strong unphysical oscillations can be present for highly anisotropic media. Enriched MPFA (EMPFA) and MPFA with full pressure support (FPS) have been proposed in the literature to reduce the strength of the oscillations. In this work, we present a simplified enhanced MPFA formulation (eMPFA) on general grids in 2D. Similar to the MPFA-O method, our formulation starts with an interaction region formed around the vertex of control volumes. Potential at the vertex is introduced as an auxiliary unknown in addition to potential at the centroids of the control volume faces. The original EMPFA and FPS solve the diffusion equation on a small volume around the vertex to close the local system of equations. To simplify the formulation, we propose an average technique to approximate potential at the vertex. Utilizing the potential value at the vertex, full potential continuity can be imposed on control volume faces to construct a more accurate flux across control volume faces. Extensive numerical experiments are conducted to test the eMPFA formulation. Specifically, we compare eMPFA with MPFA-O and two variants of EMPFA in detail. The results show that our average technique is quite robust even for highly distorted quadrilateral grids with large permeability anisotropy ratios and solutions of our formulation are in excellent agreement with EMPFA using bilinear pressure support. The new scheme also reproduces linear potential solutions and has comparable, and in some cases even better, convergence properties to those of the MPFA-O method. Finally, we generalize the formulation to include structured and unstructured triangular and polygonal grids and present the results.

CAM-SE–CSLAM: Consistent Coupling of a Conservative Semi-Lagrangian Finite-Volume Method with Spectral Element Dynamics

An algorithm to consistently couple a conservative semi-Lagrangian finite-volume transport scheme with a spectral element (SE) dynamical core is presented. The semi-Lagrangian finite-volume scheme is the Conservative Semi-Lagrangian Multitracer (CSLAM), and the SE dynamical core is the National Center for Atmospheric Research (NCAR)’s Community Atmosphere Model–Spectral Elements (CAM-SE). The primary motivation for coupling CSLAM with CAM-SE is to accelerate tracer transport for multitracer applications. The coupling algorithm result is an inherently mass-conservative, shape-preserving, and consistent (for a constant mixing ratio, the CSLAM solution reduces to the SE solution for air mass) transport that is efficient and accurate. This is achieved by first deriving formulas for diagnosing SE airmass flux through the CSLAM control volume faces. Thereafter, the upstream Lagrangian CSLAM areas are iteratively perturbed to match the diagnosed SE airmass flux, resulting in an equivalent upstream Lagrangian grid that spans the sphere without gaps or overlaps (without using an expensive search algorithm). This new CSLAM algorithm is not specific to airmass fluxes provided by CAM-SE but applies to any airmass fluxes that satisfy the Lipshitz criterion and for which the Courant number is less than one.

Numerical modeling of subsidence in saturated porous media: A mass conservative method

In this paper, a second order accurate cell-centered finite volume method (FVM) is coupled with a finite element method (FEM) to solve the deformation of a saturated porous layer based on Biot’s consolidation model. The proposed numerical technique is applied to the fully unstructured triangular grids to simulate actual geological formations. To reconstruct the pressure gradient at control volume faces, the diamond scheme is implemented as a multipoint flux approximation method. Also the least square algorithm is used to interpolate pressure at the vertices from the cell-center values. The stability of this numerical model is studied in comparison to the different FEMs through various examples. It is shown that, although the Taylor-Hood FEM has been introduced as a remedy for violation of the inf-sup condition, it does not entirely remove the non-physical oscillations. Contrary to the linear and Taylor-Hood FEMs, the proposed discretization model provides monotonic solution without imposing any restriction on the mesh or time step size. Compared to the mixed FEM, the method achieves local mass balance with fewer degrees of freedom. To couple the flow and mechanical sub-problems, the fixed-stress operator split is implemented as an iterative sequential method, due to its unconditional stability, accuracy and high rate of convergence. The accuracy of the proposed model is verified via a range of examples including analytical and numerical solutions. The performance of this methodology is assessed through modeling of subsidence in an aquifer-interbed system. This problem illustrates the capability of the model in providing stable solution in heterogeneous domains with complicated shapes.

Particle tracking approach for transport in three-dimensional discrete fracture networks

The discrete fracture network (DFN) model is a method to mimic discrete pathways for fluid flow through a fractured low-permeable rock mass, and may be combined with particle tracking simulations to address solute transport. However, experience has shown that it is challenging to obtain accurate transport results in three-dimensional DFNs because of the high computational burden and difficulty in constructing a high-quality unstructured computational mesh on simulated fractures. We present a new particle tracking capability, which is adapted to control volume (Voronoi polygons) flow solutions on unstructured grids (Delaunay triangulations) on three-dimensional DFNs. The locally mass-conserving finite-volume approach eliminates mass balance-related problems during particle tracking. The scalar fluxes calculated for each control volume face by the flow solver are used to reconstruct a Darcy velocity at each control volume centroid. The groundwater velocities can then be continuously interpolated to any point in the domain of interest. The control volumes at fracture intersections are split into four pieces, and the velocity is reconstructed independently on each piece, which results in multiple groundwater velocities at the intersection, one for each fracture on each side of the intersection line. This technique enables detailed particle transport representation through a complex DFN structure. Verified for small DFNs, the new simulation capability enables numerical experiments on advective transport in large DFNs to be performed. We demonstrate this particle transport approach on a DFN model using parameters similar to those of crystalline rock at a proposed geologic repository for spent nuclear fuel in Forsmark, Sweden.

A point-centered arbitrary Lagrangian Eulerian hydrodynamic approach for tetrahedral meshes

We present a three dimensional (3D) arbitrary Lagrangian Eulerian (ALE) hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedral meshes. The new approach stores the conserved variables (mass, momentum, and total energy) at the nodes of the mesh and solves the conservation equations on a control volume surrounding the point. This type of an approach is termed a point-centered hydrodynamic (PCH) method. The conservation equations are discretized using an edge-based finite element (FE) approach with linear basis functions. All fluxes in the new approach are calculated at the center of each tetrahedron. A multidirectional Riemann-like problem is solved at the center of the tetrahedron. The advective fluxes are calculated by solving a 1D Riemann problem on each face of the nodal control volume. A 2-stage Runge–Kutta method is used to evolve the solution forward in time, where the advective fluxes are part of the temporal integration. The mesh velocity is smoothed by solving a Laplacian equation. The details of the new ALE hydrodynamic scheme are discussed. Results from a range of numerical test problems are presented.

A Godunov-like point-centered essentially Lagrangian hydrodynamic approach

We present an essentially Lagrangian hydrodynamic scheme suitable for modeling complex compressible flows on tetrahedron meshes. The scheme reduces to a purely Lagrangian approach when the flow is linear or if the mesh size is equal to zero; as a result, we use the term essentially Lagrangian for the proposed approach. The motivation for developing a hydrodynamic method for tetrahedron meshes is because tetrahedron meshes have some advantages over other mesh topologies. Notable advantages include reduced complexity in generating conformal meshes, reduced complexity in mesh reconnection, and preserving tetrahedron cells with automatic mesh refinement. A challenge, however, is tetrahedron meshes do not correctly deform with a lower order (i.e. piecewise constant) staggered-grid hydrodynamic scheme (SGH) or with a cell-centered hydrodynamic (CCH) scheme. The SGH and CCH approaches calculate the strain via the tetrahedron, which can cause artificial stiffness on large deformation problems. To resolve the stiffness problem, we adopt the point-centered hydrodynamic approach (PCH) and calculate the evolution of the flow via an integration path around the node. The PCH approach stores the conserved variables (mass, momentum, and total energy) at the node. The evolution equations for momentum and total energy are discretized using an edge-based finite element (FE) approach with linear basis functions. A multidirectional Riemann-like problem is introduced at the center of the tetrahedron to account for discontinuities in the flow such as a shock. Conservation is enforced at each tetrahedron center. The multidimensional Riemann-like problem used here is based on Lagrangian CCH work [8,19,37,38,44] and recent Lagrangian SGH work [33–35,39,45]. In addition, an approximate 1D Riemann problem is solved on each face of the nodal control volume to advect mass, momentum, and total energy. The 1D Riemann problem produces fluxes [18] that remove a volume error in the PCH discretization. A 2-stage Runge–Kutta method is used to evolve the solution in time. The details of the new hydrodynamic scheme are discussed; likewise, results from numerical test problems are presented.

A comparison of finite difference and finite volume methods for solving the space-fractional advection-dispersion equation with variable coefficients

Transport processes within heterogeneous media may exhibit non-classical diffusion or dispersion; that is, not adequately described by the classical theory of Brownian motion and Fick's law. We consider a space fractional advection-dispersion equation based on a fractional Fick's law. The equation involves the Riemann-Liouville fractional derivative which arises from assuming that particles may make large jumps. Finite difference methods for solving this equation have been proposed by Meerschaert and Tadjeran. In the variable coefficient case, the product rule is first applied, and then the Riemann-Liouville fractional derivatives are discretised using standard and shifted Grunwald formulas, depending on the fractional order. In this work, we consider a finite volume method that deals directly with the equation in conservative form. Fractionally-shifted Grunwald formulas are used to discretise the fractional derivatives at control volume faces. We compare the two methods for several case studies from the literature, highlighting the convenience of the finite volume approach.

A hybrid pressure–density‐based algorithm for the Euler equations at all Mach number regimes

SUMMARYIn the present work, we propose a reformulation of the fluxes and interpolation calculations in the PISO method, a well‐known pressure‐correction solver. This new reformulation introduces the AUSM+ − up flux definition as a replacement for the standard Rhie and Chow method of obtaining fluxes and central interpolation of pressure at the control volume faces. This algorithm tries to compatibilize the good efficiency of a pressure based method for low Mach number applications with the advantages of AUSM+ − up at high Mach number flows. The algorithm is carefully validated using exact solutions. Results for subsonic, transonic and supersonic axisymmetric flows in a nozzle are presented and compared with exact analytical solutions. Further, we also present and discuss subsonic, transonic and supersonic results for the well known bump test‐case. The code is also benchmarked against a very tough test‐case for the supersonic and hypersonic flow over a cylinder. Copyright © 2011 John Wiley & Sons, Ltd.

A consistent and conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part III: On a staggered mesh

The consistent and conservative scheme developed on a rectangular collocated mesh [M.-J. Ni, R. Munipalli, N.B. Morley, P. Huang, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part I: on a rectangular collocated grid system, Journal of Computational Physics 227 (2007) 174–204] and on an arbitrary collocated mesh [M.-J. Ni, R. Munipalli, P. Huang, N.B. Morley, M.A. Abdou, A current density conservative scheme for incompressible MHD flows at a low magnetic Reynolds number. Part II: on an arbitrary collocated mesh, Journal of Computational Physics 227 (2007) 205–228] has been extended and specially designed for calculation of the Lorentz force on a staggered grid system (Part III) by solving the electrical potential equation for magnetohydrodynamics (MHD) at a low magnetic Reynolds number. In a staggered mesh, pressure ( p) and electrical potential ( φ) are located in the cell center, while velocities and current fluxes are located on the cell faces of a main control volume. The scheme numerically meets the physical conservation laws, charge conservation law and momentum conservation law. Physically, the Lorentz force conserves the momentum when the magnetic field is constant or spatial coordinate independent. The calculation of current density fluxes on cell faces is conducted using a scheme consistent with the discretization for solution of the electrical potential Poisson equation, which can ensure the calculated current density conserves the charge. A divergence formula of the Lorentz force is used to calculate the Lorentz force at the cell center of a main control volume, which can numerically conserve the momentum at constant or spatial coordinate independent magnetic field. The calculated cell-center Lorentz forces are then interpolated to the cell faces, which are used to obtain the corresponding velocity fluxes by solving the momentum equations. The “conservative” is an important property of the scheme, which can guarantee computational accuracy of MHD flows at high Hartmann number with a strongly non-uniform mesh employed to resolve the Hartmann layers and side layers. 2D fully developed MHD flows with analytical solutions available have been conducted to validate the scheme at a staggered mesh. 3D MHD flows, with the experimental data available, at a constant magnetic field in a rectangular duct with sudden expansion and at a varying magnetic field in a rectangular duct are conducted on a staggered mesh to verify the computational accuracy of the scheme. It is expected that the scheme for the Lorentz force can be employed together with a fully conservative scheme for the convective term and the pressure term [Y. Morinishi, T.S. Lund, O.V. Vasilyev, P. Moin, Fully conservative higher order finite difference schemes for incompressible flow, Journal of Computational Physics 143 (1998) 90–124] for direct simulation of MHD turbulence and MHD instability with good accuracy at a staggered mesh.

The Control-Volume Weighted Flux Scheme (CVWFS) for Nonlocal Diffusion and Its Relationship to Fractional Calculus

In diffusion transport, the flux at a point is typically modeled in terms of the local gradient of a potential. When heterogeneities are present, this local model can break down and it may be more appropriate to model the diffusion flux as a weighted sum of gradients present throughout the domain. Here a discrete nonlocal flux model—consistent with control-volume implementations—is developed. This scheme is referred to as the control-volume weighted flux scheme (CVWFS). The key component is the modeling of the diffusion flux at a given control-volume face in terms of a weighted sum of gradients at that face and at faces up- and downstream. Criteria for choosing the weights are proposed. This results in numerical solution schemes in which the coefficient matrix is diagonally dominant, has positive off-diagonal elements, and zero row sums. For a particular power-law weighting scheme it is shown how the CVWFS is related to the definition of the Caputo fractional derivative and the one-shift Grünwald approximation of the Riemann-Liouville fractional derivative. On developing transients and boundary condition treatments, the accuracy and suitability of the CVWFS scheme is demonstrated by solving a number of problems governed by Caputo fractional diffusion equations.

A C2-continuous control-volume technique based on cartesian grids and two-node integrated-RBF elements for second-order elliptic problems

This paper presents a new control-volume discretisation method, based on Cartesian grids and integrated-radial-basis-function elements (IRBFEs), for the solution of second-order elliptic problems in one and two dimensions. The governing equation is discretised by means of the control-volume formulation and the division of the problem domain into non-overlapping control volumes is based on a Cartesian grid. Salient features of the present method include (i) an element is defined by two adjacent nodes on a grid line, (ii) the IRBF approximations on each element are constructed using only two RBF centres (a smallest RBF set) associated with the two nodes of the element and (iii) the IRBFE solution is C2-continuous across the interface between two adjacent elements. The first feature guarantees consistency of the flux at control-volume faces. The second feature helps represent curved profiles between 2 adjacent nodes and leads to a sparse and banded system matrix, facilitating the employment of a large number of nodes. The third feature enhances the smoothness of element-based solutions, allowing a better estimation for the physical quantities involving derivatives. Numerical results indicate that (i) the proposed method can work with a wide range of the shapeparameter/RBF-width and (ii) the proposed technique yields more accurate results and faster convergence, especially for the approximation of derivatives, than the standard control-volume technique.

Simple Improvement of Momentum Interpolation Equation for Navier-Stoke Equation Solver on Unstructured Grid

Problem statement: Pressure and velocity decoupling have been source of problem in solving Navier-Stokes and continuity equation particularly in complex collocated grid. The problem of pressure velocity decoupling is usually reduced by using momentum interpolation to calculate mass flux at face of control volume. Equation of momentum interpolation was derived by assumption that the face of cell is equidistant from two neighbor cell centers and face of cell is collinear with two neighbor cell centers. This assumption is not valid in many unstructured grid and cause significant error. Approach: In this article a simple improvement of momentum interpolation for using in unstructured grid was proposed. The improvement was done by added a correction term to original equation. Results: The method was compared with original method in Kovasznay’s flow and Laminar Poisseulli Flow. The method was found able to reduce error about 40% in both cases. Conclusion: The correction added to original momentum interpolation is able to reduce error in Navier-Stokes equation solver on unstructured grid.