Cell-vertex Finite Volume Scheme Research Articles

A cell-vertex finite volume scheme for solute transport equations in open channel networks

A solute particle in a water flow behaves as a stochastic process, which is modeled by a stochastic differential equation. The solute transport equation governing macroscopic dynamics of solute concentration distribution in a locally one-dimensional open channel network is deduced from the Kolmogorov's forward equation associated to the stochastic differential equation. The cell-vertex finite volume method is applied for developing a computational scheme to numerically solve the solute transport equation. A computational domain is divided into a regular mesh, from which a dual mesh is generated. The exact solution to a local two-point boundary value problem is used for evaluating the flux at the interface of each pair of two dual cells. The scheme satisfies the total variation diminishing condition and consistently deals with any singular point such as junctions. The semi-implicit method is applied to temporal integration, and the stability condition for the time increment is presented. A series of test problems is examined in order to verify accuracy and conservative property of the scheme. Sufficiently accurate numerical solutions are obtained for test problems in a one-dimensional interval domain, while solute transport phenomena in an open channel network are correctly reproduced for cases with and without deposition of solute. It is concluded that the cell-vertex finite volume scheme is accurate, stable, and versatile in the numerical analysis of solute transport problems in open channel networks.

Excess pressure drop and drag calculations for strain-hardening fluids with mild shear-thinning: Contraction and falling sphere problems

This study extends our previous analysis on pressure-drops for strain-hardening Boger-type fluids in contraction flow settings, into those fluids that manifest mild shear-thinning properties. Numerical simulations are compared and contrasted for a variety of constitutive equations, categorised through their differences in viscometric functional response, considering application on 4:1:4 contraction–expansion flow and 2:1 flow past a sphere. Here, prior results on pressure-drop enhancement for constant shear-viscosity fluids have revealed the counter-influences of first normal stress differences and extensional viscosity. The present comparative work advances this study by selectively including the effects of shear-thinning. Suitable models to accomplish this are chosen from the class of Phan-Thien/Tanner (PTT) models, with cross-reference to FENE-models and Oldroyd-B. Furthermore, the work explores the falling sphere problem with comparison of the drag coefficient factor for various implementations. The numerical computations are performed by appealing to a well-founded hybrid finite element/finite volume algorithm, using structured triangular meshing, semi-implicit time-stepping and subcell technology. The cell-vertex finite volume scheme is particularly suited to the solution of the stress subsystem, and invokes fluctuation-distribution for upwinding and median-dual-cells for source-term representation.

Combined higher order finite volume and finite element scheme for double porosity and non-linear adsorption of transport problem in porous media

In this work, a dual porosity model of reactive solute transport in porous media is presented. This model consists of a nonlinear-degenerate advection–diffusion equation including equilibrium adsorption to the reaction combined with a first-order equation for the non-equilibrium adsorption interaction processes. The numerical scheme for solving this model involves a combined high order finite volume and finite element scheme for approximation of the advection–diffusion part and relaxation-regularized algorithm for nonlinearity-degeneracy. The combined finite volume-finite element scheme is based on a new formulation developed by Eymard et al. (2010) [10]. This formulation treats the advection and diffusion separately. The advection is approximated by a second-order local maximum principle preserving cell-vertex finite volume scheme that has been recently proposed whereas the diffusion is approximated by a finite element method. The result is a conservative, accurate and very flexible algorithm which allows the use of different mesh types such as unstructured meshes and is able to solve difficult problems. Robustness and accuracy of the method have been evaluated, particularly error analysis and the rate of convergence, by comparing the analytical and numerical solutions for first and second order upwind approaches. We also illustrate the performance of the discretization scheme through a variety of practical numerical examples. The discrete maximum principle has been proved.

The numerical prediction of planar viscoelastic contraction flows using the pom–pom model and higher-order finite volume schemes

This study investigates the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We extend our earlier work for Poiseuille flow in a planar channel and the single equation form of the extended pom–pom (SXPP) model [M. Aboubacar, J.P. Aguayo, P.M. Phillips, T.N. Phillips, H.R. Tamaddon-Jahromi, B.A. Snigerev, M.F. Webster, Modelling pom–pom type models with high-order finite volume schemes, J. Non-Newtonian Fluid Mech. 126 (2005) 207–220], to determine steady-state solutions for planar 4:1 sharp contraction flows. The numerical techniques employed are time-stepping algorithms: one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum-continuity equations by a fractional-staged Taylor–Galerkin pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. A comparison of the two finite volume approaches is presented, concentrating upon the new features posed by the pom–pom class of models in this context of non-smooth flows. Here, the dominant feature of larger shear and extension in the entry zone influences both stress and stretch, so that larger stretch develops around the re-entrant corner zone as Weissenberg number increases, whilst correspondingly stress levels decline.

A cell-vertex finite volume method for thermohydrodynamic problems in lubrication theory

In this work a coupled model in pressure and temperature is proposed and numerically solved for lubrication problems in journal-bearing devices. More precisely, a non-linear free boundary elliptic problem for the hydrodynamic behaviour, including cavitation phenomena, and a convection–diffusion problem with non-linear source due to viscous dissipation for the thermal aspects. The numerical method to calculate the pressure involves finite element approximations jointly with upwind techniques for convection and duality algorithms for non-linearities. But the main innovative goal concerns with the temperature computation by introducing a second-order cell-vertex finite volume scheme as an alternative strategy to classical first-order finite element or finite volume approximations.

Modelling pom-pom type models with high-order finite volume schemes

We are concerned with the numerical solution of viscoelastic flows using two contrasting high-order finite volume schemes. We take our earlier work for transient start-up flow in a channel and extend this beyond Oldroyd-B modelling to consider a different fluid model of the pom-pom class. This includes Single Extended form of the pom-pom model (SXPP), comparing the results of two different finite volume schemes. The numerical techniques employed are time-stepping algorithms, one of hybrid finite element/volume type, the other of pure finite volume form. The pure finite volume scheme is a staggered-grid cell-centred scheme based on area-weighting and a semi-Lagrangian formulation. This may be implemented on structured or unstructured rectangular grids, utilising backtracking along the solution characteristics in time. For the hybrid scheme, we solve the momentum/continuity equations by a fractional-staged Taylor–Galerkin/pressure-correction procedure and invoke a cell-vertex finite volume scheme for the constitutive law. This draws upon fluctuation distribution schemes (upwinding), different combinations of ‘flux’ and ‘median-dual-cell’ spatial discretisations and time-term treatments. Here, unstructured and structured meshes may be used, based largely on triangular grids. A comparison of the two finite volume approaches will be presented, concentrating upon the new features posed by the pom-pom class of models.

Resolution of pollutant concentrations in the boundary layer using a fully 3D adaptive gridding technique

This paper investigates the solution of a 3D atmospheric dispersion problem using a time-dependent adaptive gridding technique. A cell-vertex finite volume scheme based on tetrahedral elements is used to solve the atmospheric diffusion equation. Preliminary studies of dispersion from a single source in stable, unstable and neutral boundary layers have been carried out. The results show the efficiency of using adaptive grids to represent accurately the structures of plumes in the boundary layer and also the advantage of this method compared to fixed methods for mesh refinement. Some comments about the interpolation of input data such as wind fields onto unstructured meshes are also made.

Three-Dimensional Time-Marching Inviscid and Viscous Solutions for Unsteady Flows Around Vibrating Blades

A three-dimensional nonlinear time-marching method of solving the thin-layer Navier–Stokes equations in a simplified form has been developed for blade flutter calculations. The discretization of the equations is made using the cell-vertex finite volume scheme in space and the four-stage Runge–Kutta scheme in time. Calculations are carried out in a single-blade-passage domain and the phase-shifted periodic condition is implemented by using the shape correction method. The three-dimensional unsteady Euler solution is obtained at conditions of zero viscosity, and is validated against a well-established three-dimensional semi-analytical method. For viscous solutions, the time-step limitation on the explicit temporal discretization scheme is effectively relaxed by using a time-consistent two-grid time-marching technique. A transonic rotor blade passage flow (with tip-leakage) is calculated using the present three-dimensional unsteady viscous solution method. Calculated steady flow results agree well with the corresponding experiment and with other calculations. Calculated unsteady loadings due to oscillations of the rotor blades reveal some notable three-dimensional viscous flow features. The feasibility of solving the simplified thin-layer Navier–Stokes solver for oscillating blade flows at practical conditions is demonstrated.

The accuracy of cell vertex finite volume methods on quadrilateral meshes

For linear first-order hyperbolic equations in two dimensions we restate the cell vertex finite volume scheme as a finite element method. On structured meshes consisting of distorted quadrilaterals, the global error is shown to be of second order in various mesh-dependent norms, provided that the quadrilaterals are close to parallelograms in the sense that the distance between the midpoints of the diagonals is of the same order as the measure of the quadrilateral. On tensor product nonuniform meshes, the cell vertex scheme coincides with the familiar box scheme. In this case, second-order accuracy is shown without any additional assumption on the regularity of the mesh, which explains the insensitivity of the cell vertex scheme to mesh stretching in the coordinate directions, observed in practice.