Cell-centered Finite Volume Method Research Articles

Coupled problem of the inverse design and constraint optimization

Two dimensional inviscid compressible flow solver has been implemented, validated and extended for completing inverse design based optimization problems. Euler equations are used for modelling basic physics. The standard cell centred finite volume method has been applied with Roe’s approximated Riemann solver, MUSCL (Monotone Upstream Schemes for Conservation Laws) approach and Mulder limiter. Stratford’s separation prediction method with constrained SQP (Sequential Quadratic Programming) optimization is implemented to evaluate the optimum pressure distribution at strong adverse pressure gradient flow conditions. A particular inverse design method is developed to complete wall surface modification till the optimum target pressure distribution is reached by means of inverse, wall modification and direct algorithms. The direct solver is validated and the coupled system of the inverse design method with constraint optimization is tested over the NACA 65-410 airfoil.

A totally Eulerian finite volume solver for multi-material fluid flows: Enhanced Natural Interface Positioning (ENIP)

This work concerns the simulation of compressible multi-material fluid flows and follows the method FVCF-NIP described in the former paper (Braeunig et al., 2009 [7]). This cell-centered finite volume method is totally Eulerian since the mesh is not moving and a sharp interface, separating two materials, evolves through the grid. A sliding boundary condition is enforced at the interface, and mass, momentum and total energy are conserved. Although this former method performs well on 1D test cases, the interface reconstruction suffers of poor accuracy in conserving shapes for instance in linear advection. This situation leads to spurious instabilities of the interface. The Enhanced-NIP method presented in the present paper provides a solution to an inconsistency in the former NIP method. This solution strikingly improves the numerical results. It takes advantage of a more consistent description of the interface in the numerical scheme. Results for linear advection and compressible Euler equations for inviscid fluids are presented to assess the benefits of this new method.

Numerical and Experimental Flow Analysis of the Wang-Zwische Double-Lumen Cannula

An experimental and numerical analysis was performed for the Wang-Zwische double-lumen cannula (DLC) (Avalon Elite). The aim of this work was to provide insight for future improvement by characterizing the fluid dynamic behavior of the novel catheter with metrics often associated with blood trauma. Pressure and flow distributions were measured on a steady-flow rig using a 50% glycerol-water mixture by imposing a 2 L/min flow rate across the drainage and infusion lumens. The fluid was modeled as Newtonian with density of 1050 kg/m³ and dynamic viscosity of 0.0035 kg/m·s. Reynolds numbers typical for transitional flow (Re = 2000-2500) were computed within the lumens because of the changing cross-sections of the cannula geometry. Numerical computations were performed using the steady three-dimensional Reynolds-averaged Navier-Stokes (RANS) equations and the low-Reynolds k-ω turbulence model. Discretization of governing equations was based on a cell-centered finite volume method. Numerical results correlated well with global performance of the cannula, allowing evaluation of the geometry toward potential blood trauma. Peak wall shear stress (WSS) in the drainage lumen was higher than that of infusion lumen, mainly due to the presence of side holes. Furthermore, recirculation regions were predicted in transition tubing to connectors of both the drainage and the infusion lumens because of adverse pressure gradients caused by the sudden enlargement of the cannula geometry. In this three-dimensional computational fluid dynamics (CFD) study, we observed higher peak WSS values for the drainage lumen, which may potentially cause blood trauma. Furthermore, recirculation regions were predicted in the proximity of the exit sections of both the infusion and drainage lumens, which may contribute to thrombosis formation. This study provides insight for future DLC modifications in minimizing cannula-induced blood trauma and thrombogenicity in long-term applications.

Hybrid finite element/volume method for 3D incompressible flows with heat transfer

An implicit hybrid finite element (FE)/volume solver has been extended to incompressible flows coupled with the energy equation. The solver is based on the segregated pressure correction or projection method on staggered unstructured hybrid meshes. An intermediate velocity field is first obtained by solving the momentum equations with the matrix-free implicit cell-centred finite volume (FV) method. The pressure Poisson equation is solved by the node-based Galerkin FE method for an auxiliary variable. The auxiliary variable is used to update the velocity field and the pressure field. The pressure field is carefully updated by taking into account the velocity divergence field. Our current staggered-mesh scheme is distinct from other conventional ones in that we store the velocity components at cell centres and the auxiliary variable at vertices. The Generalized Minimal Residual (GMRES) matrix-free strategy is adapted to solve the governing equations in both FE and FV methods. The presented 2D and 3D numerical examples show the robustness and accuracy of the numerical method.

2차원 및 축대칭 운동체 주위의 초공동 현상에 대한 수치해석

Super-cavitating flows around under-water bodies are being studied for drag reduction and dramatic speed increase. In this paper, high speed super-cavitating flow around a two-dimensional symmetric wedge-shaped body were studied using an unsteady Reynolds-averaged Navier-Stokes equations solver based on a cell-centered finite volume method. To verify the computational method, flow over a hemispherical head-form body was simulated and validated against existing experimental data. Various computational conditions, such as different wedge angles and caviation numbers, were considered for the super-cavitating flow around the wedge-shaped body. Super-cavity begins to form in the low pressure region and propagates along the wedge body. The computed cavity lengths and velocities on the cavity boundary with varying cavitation number were validated by comparing with analytic solution.

Type-VI and Type-V Shock-Shock Interactions on Double-Wedge Geometries Using AUSM+ on Unstructured Grid

The Euler equations are solved on unstructured triangular meshes for hypersonic flow over double-wedge geometries. The driving algorithm is an upwind biased cell centered finite volume method. AUSM+ method is used to split the fluxes. Edney (1968) studied the shock interactions by impinging an externally generated planar oblique shock on the bow shock generated by a cylinder. Depending upon the parametric conditions Edney classified the shock interactions in different types. Two interaction topologies, namely Type-VI and Type-V and the transition from Type-VI to Type-V are studied in details. Both six-shock and seven-shock configurations of Type-V interaction are presented.

On the coupling of finite volume and discontinuous Galerkin method for elliptic problems

The coupling of cell-centered finite volume method with primal discontinuous Galerkin method is introduced in this paper for elliptic problems. Convergence of the method with respect to the mesh size is proved. Numerical examples confirm the theoretical rates of convergence. Advantages of the coupled scheme are shown for problems with discontinuous coefficients or anisotropic diffusion matrix.

Guaranteed and Fully Robust a posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients

We study in this paper a posteriori error estimates for H 1-conforming numerical approximations of diffusion problems with a diffusion coefficient piecewise constant on the mesh cells but arbitrarily discontinuous across the interfaces between the cells. Our estimates give a global upper bound on the error measured either as the energy norm of the difference between the exact and approximate solutions, or as a dual norm of the residual. They are guaranteed, meaning that they feature no undetermined constants. (Local) lower bounds for the error are also derived. Herein, only generic constants independent of the diffusion coefficient appear, whence our estimates are fully robust with respect to the jumps in the diffusion coefficient. In particular, no condition on the diffusion coefficient like its monotonous increasing along paths around mesh vertices is imposed, whence the present results also include the cases with singular solutions. For the energy error setting, the key requirement turns out to be that the diffusion coefficient is piecewise constant on dual cells associated with the vertices of an original simplicial mesh and that harmonic averaging is used in the scheme. This is the usual case, e.g., for the cell-centered finite volume method, included in our analysis as well as the vertex-centered finite volume, finite difference, and continuous piecewise affine finite element ones. For the dual norm setting, no such a requirement is necessary. Our estimates are based on H(div)-conforming flux reconstruction obtained thanks to the local conservativity of all the studied methods on the dual grids, which we recall in the paper; mutual relations between the different methods are also recalled. Numerical experiments are presented in confirmation of the guaranteed upper bound, full robustness, and excellent efficiency of the derived estimators.

Numerically analyzing more efficiently high-lift aerodynamics of wing/body model with omni-tree Cartesian grids

A modified Cartesian grid method is introduced to simulate the flow fields around wing/body high-lift configurations by using a multi-zone technique. With a face-to-face algorithm, the Euler equations are solved numerically, in combination with the cell-center finite volume method and dual-time stepping scheme. The multi-zone technique is employed to simplify the great difficulties in grid generation. The face-to-face algorithm is constructed to precisely establish the information exchange of flow fields on the interface between zones. The numerical results are presented and compared with experimental data.

Cell-Centered Finite-Volume Time-Domain Method for Conducting Media

This letter describes a cell-centered finite-volume time-domain method for solving electromagnetic problems which involve media having finite conductivity. Two different approaches, the additive induced source technique and the fractional-step technique (FST), are considered and compared with a reliable numerical method. Namely, the former treats the induced electric current density as an additive source term, whereas the latter is based on the FST.

Local pressure preconditioning method for steady incompressible flows

The convergence and accuracy characteristics of the preconditioned incompressible Euler and Navier–Stokes equations are studied. An object-oriented C++ numerical code has been developed for solving the inviscid and viscous, steady, incompressible flows problems. The code is based on the cell-centred finite volume method. In this scheme, two-dimensional incompressible Euler and Navier–Stokes equations are modified by a robust artificial compressibility (AC) and a local preconditioning matrix of pressure-sensor type. The preconditioned equations are solved with the Jameson's numerical approach, i.e. artificial dissipation and artificial viscosity terms under the form of a fourth- and second-order derivative, respectively. An explicit four-stage Runge–Kutta integration algorithm is applied to obtain the steady-state condition. The computed results include the steady-state solution of flow past the NACA-hydrofoils and a circular cylinder in free stream, for which the numerical results are compared with numerical works of other researchers. Good agreement is observed. The effects of AC parameter, artificial viscosity and dissipation factor, and local preconditioning coefficient on convergence rate and solution accuracy are tested by computing flow over the NACA0012 hydrofoil. In addition, some important design criteria of a preconditioner, such as stiffness reduction, hyperbolicity, symmetrisability, accuracy preservation for M → 0, and M-property have been examined analytically.

Numerical analysis of unsteady cavitating flow in an axial inducer

This article presents a numerical simulation of an unsteady cavitating flow. First, the analysis was carried out for a two-dimensional (2D) venturi-type geometry, and then for a 2D blade cascade simulating an axial inducer. The main purpose of this work is to explore the local cavitation instabilities, such as alternate blade cavitation and rotating blade cavitation, which can appear in axial inducers. The numerical simulation was performed using a commercial code based on a cell-centred finite-volume method. The cavitation model used for calculations assumes a thermal equilibrium between phases. It is based on the classical conservation equations of the vapour phase and the mixture phase, with mass transfer owing to the cavitation appearing as a source and a sink term in the vapour mass fraction equation. The mass transfer rate is derived from a simplified Rayleigh—Plesset model for bubble dynamics.

Implementation of a second-order upwind method in a semi-implicit two-phase flow code on unstructured meshes

A two-phase flow analysis code, CUPID, has been developed for a realistic simulation of thermal–hydraulic phenomena in nuclear reactor components. In the CUPID code, a two-fluid three-field model is adopted and the governing equations are solved on unstructured meshes. To obtain the numerical solution, the semi-implicit method of the RELAP5 code was used with some modifications for a cell-centered finite volume method. In this work, a second-order upwind method was implemented for the convective terms of the CUPID code. To get the slopes of the convective quantities, we adopted the Frink’s reconstruction method (1994, AIAA Paper 94-0061) and modified it for an application to arbitrary polyhedral cells. To stabilize the numerical solutions, the Barth and Jesperson’s slope limiter was used. In order to evaluate the enhanced accuracy and ensure the robustness of the implemented scheme, numerical tests were performed using conceptual single- and two-phase flow problems, which include a strong phase change and very heterogeneous phase distributions.

New Formulation for Stress Calculation: Application to Viscoelastic Flow in a T-Junction

In this study, a new formulation is proposed for the calculation of stress components at control-volume faces, within the context of cell-centered finite-volume methods which have the stress tensor as one of the main dependent variables. Test case results from calculations with viscoelastic fluids flowing through a T-junction demonstrate the merits of the method. Previous formulations for stress interpolation yielded results that would depend on the time-step value employed, even when calculating steady-state problems. We have removed this inconsistency by devising an improved method that gives results independent of the time step and that in addition is more robust, with a wider range of allowable Deborah numbers. A FENE-CR constitutive model is used to replicate the known viscoelastic nature of blood, and results are given for varying fluid elasticities, at values of Reynolds number and extraction ratio typical of hemodynamic applications.

Monoslope and multislope MUSCL methods for unstructured meshes

We present new MUSCL techniques associated with cell-centered finite volume method on triangular meshes. The first reconstruction consists in calculating one vectorial slope per control volume by a minimization procedure with respect to a prescribed stability condition. The second technique we propose is based on the computation of three scalar slopes per triangle (one per edge) still respecting some stability condition. The resulting algorithm provides a very simple scheme which is extensible to higher dimensional problems. Numerical approximations have been performed to obtain the convergence order for the advection scalar problem whereas we treat a nonlinear vectorial example, namely the Euler system, to show the capacity of the new MUSCL technique to deal with more complex situations.

Thermal Behaviour of Steel Plate during Accelerated Cooling

Recent trends in the production of high strength steel plate call for increasingly sophisticated thermo-mechanical treatment schedules, including the use of high rate accelerated cooling after finish rolling in order to achieve the desired microstructure and mechanical properties. Achieving the necessary cooling process control accuracy in such cases requires a sound understanding and description of the interactions between external heat transfer processes and changes in internal energy due to phenomena such as solid-state phase transformations. The thermal physical properties of the evolving microstructures of complex phase and martensitic steels vary greatly, and are strongly dependent on temperature and constituent phases. As a result, critical parameters such as thermal diffusivity cannot be accurately estimated without appropriate linkage to both phase transformation kinetics and temperature. In the present study, a numerical simulation has been developed to investigate the unsteady heat transfer and phase transformation behaviour of a moving steel plate during accelerated cooling. The simulation includes semi-empirical microstructure evolution sub-models, fitted to measured CCT data using non-linear regression. These are coupled to thermal-physical properties sub-models and thermal conduction calculations. A comprehensive suite of thermal boundary condition models which account for direct water cooling, forced convection film boiling, air cooling, radiation and heat transfer between plate and transport rollers are also included. The required equations for the plate temperature and microstructure evolution are solved numerically using a cell centred finite volume method, and the model has been validated by comparing simulated cooling stop temperatures with measurements obtained on the plate cooling section of an industrial plate mill. The predicted cooling stop temperatures of steel plates for different thicknesses, velocities and water flow rates are in good agreement with plant operational data.

A posteriori error analysis of a cell-centered finite volume method for semilinear elliptic problems

In this paper, we conduct a goal-oriented a posteriori analysis for the error in a quantity of interest computed from a cell-centered finite volume scheme for a semilinear elliptic problem. The a posteriori error analysis is based on variational analysis, residual errors and the adjoint problem. To carry out the analysis, we use an equivalence between the cell-centered finite volume scheme and a mixed finite element method with special choice of quadrature.

A hybrid finite volume/finite element method for incompressible generalized Newtonian fluid flows on unstructured triangular meshes

This paper presents a hybrid finite volume/finite element method for the incompressible generalized Newtonian fluid flow (Power-Law model). The collocated (i.e. non-staggered) arrangement of variables is used on the unstructured triangular grids, and a fractional step projection method is applied for the velocity-pressure coupling. The cell-centered finite volume method is employed to discretize the momentum equation and the vertex-based finite element for the pressure Poisson equation. The momentum interpolation method is used to suppress unphysical pressure wiggles. Numerical experiments demonstrate that the current hybrid scheme has second order accuracy in both space and time. Results on flows in the lid-driven cavity and between parallel walls for Newtonian and Power-Law models are also in good agreement with the published solutions.

A multigrid method for the Poisson–Nernst–Planck equations

A computational technique for solving the Poisson–Nernst–Planck (PNP) equations is developed which overcomes the poor convergence rates of commonly used algorithms. The coupled Poisson and charge continuity equations are discretized using an unstructured cell-centered finite volume method. A Newton–Raphson linearization accounting for the coupling between the equations through boundary conditions, and the space charge and drift terms, is developed. The resulting linear system of equations is solved using an algebraic multigrid method, with coarse level systems being created by agglomerating finer-level equations based on the largest coefficients of the Poisson equation. A block Gauss–Seidel update is used as the relaxation method. The method is shown to perform well for the transport of K + and Cl - in a synthetic ion channel for driving voltages, surface charges, ion concentrations and channel aspect ratios ranging over several orders of magnitude.

A Conservative Coupled Flow/Transport Model with Zero Mass Error

A fully conservative form applied to a coupled system of two-dimensional water flow and solute motion is presented. A cell-centred finite volume method based on Roe's approximate Riemann solver with unstructured grids is formulated. The bed slope source terms are discretized following an upwind approach and a semi-implicit treatment is used for the friction source terms. The centered discretization of the diffusion terms is in an implicit way. It is shown that this numerical technique reproduces almost exactly the steady state of still water and enables to achieve zero numerical errors in unsteady flow over configurations with strong variations on bed slope. The model ensures a global conservation and positive values of both water level and solute concentration. Numerical results show the effectiveness of the model in solute transport over real complex geometries.