Cell-centered Finite Volume Method Research Articles

A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

AbstractWe propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010

Two-dimensional finite volume method for dam-break flow simulation

A numerical model based upon a second-order upwind cell-center finite volume method on unstructured triangular grids is developed for solving shallow water equations. The assumption of a small depth downstream instead of a dry bed situation changes the wave structure and the propagation speed of the front which leads to incorrect results. The use of Harten-Lax-vau Leer (HLL) allows handling of wet/dry treatment. By usage of the HLL approximate Riemann solver, also it make possible to handle discontinuous solutions. As the assumption of a very small depth downstream of the dam can change the nature of the dam break flow problem which leads to incorrect results, the HLL approximate Riemann solver is used for the computation of inviscid flux functions, which makes it possible to handle discontinuous solutions. A multidimensional slope-limiting technique is applied to achieve second-order spatial accuracy and to prevent spurious oscillations. To alleviate the problems associated with numerical instabilities due to small water depths near a wet/dry boundary, the friction source terms are treated in a fully implicit way. A third-order Runge–Kutta method is used for the time integration of semi-discrete equations. The developed numerical model has been applied to several test cases as well as to real flows. The tests are tested in two cases: oblique hydraulic jump and experimental dam break in converging-diverging flume. Numerical tests proved the robustness and accuracy of the model. The model has been applied for simulation of dam break analysis of Torogh in Iran. And finally the results have been used in preparing EAP (Emergency Action Plan).

Cell-centered finite volume methods with flexible stencils for diffusion equations on general nonconforming meshes

A cell-centered finite volume method is presented for discretizing diffusion operator on general nonconforming meshes. The node values are accurately approximated using a new weighted interpolation formula, in which the calculation of the weight is adaptive to both geometric parameters and diffusion coefficients. It follows that an explicit expression, composed of cell-centered unknowns only, is obtained for the discretization of normal flux. Numerical results demonstrate that linear solutions are reproduced exactly on the nonconforming random grids, and that the convergence rate is close to second order for non-linear or discontinuous problems.

Non-Newtonian effects of blood flow in complete coronary and femoral bypasses

A numerical investigation of non-Newtonian steady blood flow in a complete idealized 3D bypass model with occluded native artery is presented in order to study the non-Newtonian effects for two different sets of physiological parameters (artery diameter and inlet Reynolds number), which correspond to average coronary and femoral native arteries. Considering the blood to be a generalized Newtonian fluid, the shear-dependent viscosity is evaluated using the Carreau–Yasuda model. All numerical simulations are performed by an incompressible Navier–Stokes solver developed by the authors, which is based on the pseudo-compressibility approach and the cell-centred finite volume method defined on unstructured hexahedral computational grid. For the time integration, the fourth-stage Runge–Kutta algorithm is used. The analysis of numerical results obtained for the non-Newtonian and Newtonian flows through the coronary and femoral bypasses is focused on the distribution of velocity and wall shear stress in the entire length of the computational model, which consists of the proximal and distal native artery and the connected end-to-side bypass graft.

AMG preconditioning for sedimentary basin simulations in Temis calculator

Basin modelling aims at reconstructing the time evolution of a sedimentary basin in order to make quantitative predictions of geological phenomena leading to hydrocarbon accumulations. It accounts for porous medium compaction, heat transfer, hydrocarbon generation and migration. These physical phenomena are modelled by partial differential equations [Schneider, F., Wolf, S., Faille, I., Pot, D. 2000. A 3D basin model for hydrocarbon potential evaluation: application to Congo offshore. Oil and Gas Science Technologie, Rev-IFP 55, 3–13] representing the mass balances of solid, fluids (water and oil) coupled with Darcy's law and a compaction law. These equations are discretized using a cell-centered Finite Volume method in space and an implicit Euler integration in time. At each time step, the resulting nonlinear system is solved using Newton's method ending up at each Newton iteration with the solution of a linear system which represents the most cpu-time-consuming part of the basin simulator. Solving these ill-conditioned linear systems by an iterative Krylov algorithm is a challenging task. It requires the design of efficient preconditioners which should be scalable with respect to the size of the mesh, the size of the heterogeneities, as well as the number of processors on distributed parallel computing architectures. They should in addition efficiently cope with the various types of unknowns and their coupling in the linear system, namely the pressure, the saturation, the hydrocarbon compositions, and the geostatic load. This paper focuses on the CPR-AMG preconditioner introduced in Lacroix, S., Vassilevski, Y., Wheeler M.F. [2001. Iterative solvers of the implicit parallel accurate reservoir simulator (IPARS). Numerical Linear Algebra with Applications, 537–549] based on the earlier works of Behie, A., Vinsome, P.K.W. [1982. Block iterative methods for fully implicite reservoir simulation. Society of Petroleum Engineers 22, 658–668]; Wallis, J. [1985. Constrained Residual Acceleration of Conjugate Residual Methods. 8 SPE paper 13536, Dallas, Texas], and successfully applied in reservoir simulation [Gratien, J.M., Magras, J.F., Quandalle, P., Ricois, O., 2004. Introducing a new generation of reservoir simulation software. In: ECMOR IX Proceedings, Cannes; Achdou, Y., Bonneau, P., Masson, R., Quandalle, P., 2006. Algebraic multigrid methods and block preconditioning for mixed elliptic hyperbolic linear systems. In: ECMOR X Proceedings, Amsterdam], as well as in basin modelling [Scheichl, R., Masson, R., Wendebourg, J., 2003. Decoupling and block preconditioning for sedimentary basin simulations. Computational Geosciences 7 (4), 295–318; Masson, R., Quandalle, P., Requena, S., Scheichl, R., 2004. Parallel preconditioning for sedimentary basin simulations. Lecture notes in computer science 2907. In: Proceedings of the Fourth International Conference on Large Scale Scientific Computations. Springer, Sozopol, Bulgaria]. In this article, this approach is extended to compositional basin models, its performance is assessed on real challenging test cases on distributed parallel architectures, and compared with classical preconditioners.

A second-order accurate finite volume method for the computation of electrical conditions inside a wire-plate electrostatic precipitator on unstructured meshes

In this paper, an unstructured cell-centered second-order accurate finite volume method is presented for the computation of electrical conditions inside wire-plate electrostatic precipitators. The potential equation was discretized using a second-order accurate scheme by invoking a new type of special line-structure. The space–charge density equation was discretized using a second-order upwind scheme, and solved using a new direct method. The local gradients are reconstructed by a weighted least-square reconstruction method. The method can deal with complex geometries by using unstructured meshes. Numerical experiments show that the predicted results agree well with the existing experimental data.

Compressible viscous flow solver

Nowadays, in spite of disadvantages of turbulence closure models for RANS (Reynolds Averaged Navier-Stokes equations), they are at present the only tools available for the computation of complex turbulent flows of practical relevance. Their popularity comes from high efficiency in terms of accuracy and computational cost, which makes them widely used in commercial codes and related multidisciplinary applications. Hence, for modelling compressible flow, as a framework of complex inverse design optimisation tool, Navier-Stokes solver is implemented by using k- ω turbulence model in C++ environment. The governing equations in conservative form are deduced by using Favre averaging to filter local fluctuations. The code is based on structured, density based cell centred finite volume method. The convective terms are discretized by Roe approximated Riemann method. Central discretization is applied for diffusive terms. MUSCL approach is implemented for higher order spatial reconstruction with Mulder limiter for monotonicity preserving. Wilcox k- ω two equations turbulence model is implemented for turbulence modelling. The explicit system of the equations is solved by the 4 th order Runge-Kutta method. The numerical boundary conditions are based on the method of characteristics. The interest is mostly in high speed aeronautical applications with the possibility of extension for surface optimisation. Hence, the applied validational test cases are in transonic and supersonic flow regime: circular bump in the transonic channel and compression corner.

A Posteriori Error Estimate and Adaptive Mesh Refinement for the Cell-Centered Finite Volume Method for Elliptic Boundary Value Problems

We extend a result of Nicaise [SIAM J. Numer. Anal., 43 (2005), pp. 1481-1503] for the a posteriori error estimation of the cell-centered finite volume method for the numerical solution of elliptic problems. Having computed the piecewise constant finite volume solution $u_h$, we compute a Morley-type interpolant $\mathcal{I} u_h$. For the exact solution $u$, the energy error $\norm{\nabla_{\mathcal{T}}(u-\mathcal{I} u_h)}{L^2}$ can be controlled efficiently and reliably by a residual-based a posteriori error estimator $\eta$. The local contributions of $\eta$ are used to steer an adaptive mesh-refining algorithm. A model example serves the Laplace equation in two dimensions with mixed Dirichlet-Neumann boundary conditions.

A monotone nonlinear finite volume method for diffusion equations on conformal polyhedral meshes

We have developed a new monotone cell-centered finite volume method for the discretization of diffusion equations on conformal polyhedral meshes. The proposed method is based on a nonlinear two-point flux approximation. For problems with smooth diffusion tensors and Dirichlet boundary conditions the method is interpolation-free. An adaptive interpolation is applied on faces where diffusion tensor jumps or Neumann boundary conditions are imposed. The interpolation is based on physical relationships such as continuity of the diffusion flux. The second-order convergence rate is verified with numerical experiments.

제트 노즐의 배치가 콴다 날개의 성능에 미치는 영향

The Coanda effects demonstrate that a jet stream applied tangential to a curved surface can generate lift force by increasing the circulation. Many experimental and numerical studies have been performed on the Coanda effect and it is found to be useful in various fields of aerodynamics. The Coanda effect may have practical application to marine hydrodynamics since various control surfaces are being used to control behaviors of ships and offshore structures. In the present study, numerical computations are performed to find the applicability of the Coanda effect to the marine control surfaces. For the purpose, changes in flow characteristics around a flapped foil due to the Coanda effect have been simulated by RANS equations discretized with a cell-centered finite volume method (FVM). In the process, special attention has been given to the influence of jet nozzle arrangement on the lift characteristics of the Coanda foil. It is found that the shape as well as the location of the jet intake and jet exit affects the lift performance of the foil significantly.

Numerical Solution of Newtonian and Non‐Newtonian Flows in Bypass

AbstractThis paper deals with a numerical solution of laminar incompressible steady flows of Newtonian and non–Newtonian fluids through bypass of a restricted vessel. Blood flow is considered to be Newtonian in the case of vessels of large diameters as aorta. On the other hand, with decreasing diameter of a vessel the non–Newtonian behavior of blood can play a significant role. One could describe these problems using Navier–Stokes equations and continuity equation as a model. In the case of Newtonian fluids one considers constant viscosity compared to non–Newtonian fluids where viscosity varies and can depend on the tensor of deformation. In order to find numerical solution, the system of equations is completed using an artificial compressibility method. The space derivatives are discretised using a cell centered finite volume method and arising system of ordinary differential equations is solved using an explicit multistage Runge–Kutta method with given steady boundary conditions. The steady solution is achieved for time t→∞ and steady boundary conditions. The results can be used in the field of cardiovascular research. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical solution of turbulent jet flows

AbstractThe work deals with numerical modelling of several turbulent 3D jet flows: steady impinging jet, steady free jet in cross–flow, synthetic free jet (unsteady) and synthetic impinging jet (unsteady). The numerical method is based on artificial compressibility method with dual time extension for unsteady cases. Space discretization uses cell–centered finite volume method with third order accurate upwind approximation for convection, the time discretisations are implicit. Turbulence is modelled using two–equation eddy viscosity models and by explicit algebraic Reynolds stress model (EARSM by Wallin and Hellsten). The results of first three cases are compared with measurements. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A Finite-Volume Method for Navier-Stokes Equations on Unstructured Meshes

A novel finite-volume formulation is proposed for unsteady solutions on complex geometries. A computer code based on a cell-centered finite-volume method is developed to solve both two-dimensional (2-D) and three-dimensional (3-D) Navier-Stokes equations for incompressible laminar flow on unstructured grids. A collocated (i.e., nonstaggered) arrangement of variables is used. The convective terms have provision for a variable upwinding factor, and the diffusion fluxes are computed in a novel and natural way. The pressure–velocity decoupling is avoided by momentum interpolation. The method is shown to have nearly second-order accuracy even on nonorthogonal grids. Some Navier-Stokes solutions, both 2-D and 3-D, are presented to verify the method with standard benchmark solutions. The comparison of present results with those in the literature is good. A computational study of 2-D laminar flow and heat transfer past a triangular cylinder in free stream is presented for the range 10 ≤ Re ≤ 200.

Study of a flexible disk rotating close to a rigid rotating wall considering fluid inertia effects

The present study is a numerical simulation about the dynamics of a flexible disk coupled to thin air film and rotating close to a rigid rotating wall. The idea of a flexible disk rotating in a close proximity of a rigid rotating wall is introduced and studied with two new types of flat stabilizers, co-rotating and counter-rotating flat stabilizers, besides the well-known fixed-stabilizer type which has been studied extensively in earlier works. In the present study, the flexible disk is modeled using linear plate theory and the air flow between the flexible disk and the rigid wall is modeled using Navier–Stokes and continuity equations. The flow equations are discretized using cell centered finite volume method (FVM) and solved numerically with the SIMPLE algorithm, while the spatial terms in the disk model are discretized using finite difference method (FDM) and time integration is performed using fourth-order Runge–Kutta method. The effect of inertia and coriollis forces on the disk displacement and air-film pressure is studied, also the dependence of these forces on the rotation speed, initial gap size and inlet-hole radius is investigated. A transient numerical code is developed to compare the stability boundaries for the different types of flat stabilizer at a wide range of circumferential mode numbers. The numerical results showed an improved stability of the flexible disk when rotating close to a counter-rotating flat stabilizer compared with co-rotating and fixed flat stabilizers.

The Jameson’s numerical method for solving the incompressible viscous and inviscid flows by means of artificial compressibility and preconditioning method

A computational code is developed using cell-centered finite volume method with a non-uniform grid for solving the incompressible viscous and inviscid flows. The method has been used to determine the steady incompressible inviscid flows past a cylinder in free stream, the steady incompressible inviscid flows past a circular bump through a channel, and also the steady incompressible viscous flows past a backward facing-step. In this method, the 2D Navier–Stokes equations (or 2D incompressible Euler equations for inviscid flow), which are modified by artificial compressibility and preconditioning concepts, are solved with the Jameson’s artificial dissipation and viscosity terms under the form of a fourth- and second-order x-derivative, respectively. An explicit fourth-order Runge–Kutta integration algorithm is applied to find the steady state condition. The effects of CFL number, artificial viscosity coefficient, and pseudo-compressibility parameter in convergence of solution are investigated.

A direct design approach using the Euler equations

Recently, a fully coupled formulation of the surface shape design problem, called the direct design approach, has been proposed in which both the target surface pressure and the unknown nodal coordinates appear explicitly in the formulations. The proposed method is generally applicable, but in the past it has only been applied in the context of ideal fluid flows [Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2003, Direct design of ducts. Journal of Fluids Engineering, Transaction ASME, 125, 158–165; Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2002, Direct design of shape. Numerical Heat Transfer-Part B, 41, 501–510; Ashrafizadeh, A., Raithby, G.D. and Stubley, G.D. 2004, Direct design of airfoil shape with a prescribed surface pressure. Numerical Heat Transfer, Part B: Fundamentals, 46(6), 505–527]. The present article extends the application of this method to the design of ducts carrying flows governed by the nonlinear coupled Euler equations, using a cell-centered finite volume method to discretize the governing equations. The details of the linearization and the imposition of the target pressure are discussed in this article. Also, the validation test cases and the design examples are presented, which show the robustness of the method in handling complex geometries and complex physical phenomena (such as shock waves). It is also shown that the computational cost of a direct shape design solution is comparable to the solution of the corresponding analysis problem.

Parallel solver for the three-dimensional Cartesian-grid-based time-dependent Schrödinger equation and its applications in laser-H2+interaction studies

A parallelized three-dimensional Cartesian-grid-based time-dependent Schr\"odinger equation (TDSE) solver for molecules with a single electron, assuming the motion of the nucleus is frozen, is presented in this paper. An explicit stagger-time algorithm is employed for time integration of the TDSE, in which the real and imaginary parts of the wave function are defined at alternate times, while a cell-centered finite-volume method is utilized for spatial discretization of the TDSE on Cartesian grids. The TDSE solver is then parallelized using the domain decomposition method on distributed memory machines by applying a multilevel graph-partitioning technique. The solver is validated using a $\mathrm{H}_{2}{}^{+}$ molecule system, both by observing the total electron probability and total energy conservation without laser interaction, and by comparing the ionization rates with previous two-dimensional axisymmetric simulation results with an aligned incident laser pulse. The parallel efficiency of this TDSE solver is presented and discussed; the parallel efficiency can be as high as 75% using 128 processors. Finally, examples of the temporal evolution of the probability distribution of laser incidence onto a $\mathrm{H}_{2}{}^{+}$ molecule at inter-nuclear distance of $9\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ ($\ensuremath{\chi}=0\ifmmode^\circ\else\textdegree\fi{}$ and 90\ifmmode^\circ\else\textdegree\fi{}) and the spectral intensities of harmonic generation at internuclear distance of $2\phantom{\rule{0.3em}{0ex}}\mathrm{a.u.}$ ($\ensuremath{\chi}=0\ifmmode^\circ\else\textdegree\fi{}$, 30\ifmmode^\circ\else\textdegree\fi{}, 60\ifmmode^\circ\else\textdegree\fi{}, and 90\ifmmode^\circ\else\textdegree\fi{}) are presented to demonstrate the powerful capability of the current TDSE solver. Future possible extensions of the present method are also outlined at the end of this paper.

Simulation of Flow Separation at the Wing-Body Junction with Different Fairings

[Abstract] The numerical simulation and analyses of the flow separation at the wingbody junction with different designed fairings for a transporter are carried on by using a viscous flow solver based on hybrid unstructured meshes in this paper. In the process of generating hybrid meshes, Pirzadeh’s advancing-layer method is improved to generate high quality hybrid prismatic/pyramid/tetrahedral unstructured meshes in the boundary-layer region, and advancing-front method is used to construct isotropic tetrahedral meshes in the residual flow region. Navier-Stokes equations are solved using a cell-centered finite-volume method with Spalart-Allmaras one-equation turbulence model. And according to Bettina and Tomaro’s theory, time is marched by means of an implicit Gauss-Seidel relaxation procedure, which is constructed by using the first-order linearizing of flux vector and the maximal eigenvalue splitting of flux Jacobian matrix. Withal local time stepping and implicit residual smoothing artifice are used to accelerate convergence. The consistence of the computational results and the experimental measurements proves the superior capability of simulating wing-body interferences on a transporter by using a viscous flow solver based on hybrid unstructured meshes. In addition, numerical studies indicate that wing-body fairing could allay flow separation at the wing-body junction, increase the lift-drag ratio and improve the aerodynamic performance.

Effects of stent porosity on hemodynamics in a sidewall aneurysm model

Computation and experiment have been complementarily performed to study the fluid flow inside a stented lateral aneurysm anchored on the straight parent vessel. The implicit solver was based on the time-dependent incompressible Navier–Stokes equations of laminar flow. Solutions were generated by a cell-center finite-volume method that used second-order upwind and second-order center flux difference splitting for the convection and diffusion term, respectively. The second-order Crank–Nicolson method was used in the time integration term. Experimental techniques used were flow visualization (FV) and particle tracking velocimetry (PTV). Experimentally, the straight afferent vessel had an inner diameter 10 mm. The diameters of the aneurysmal orifice, neck, and fundus were 14, 10, and 15 mm, respectively, and the distance between the orifice and dome measured 20 mm. A 30 mm long helix-shaped stent was tested. Four stent porosities of 100%, 70%, 50%, and 25% were examined. Volume-flow rate waveform of a cerebral artery was considered with a maximum Reynolds number of 250 and Womersley number of 3.9. Results are presented in terms of the pulsatile main and secondary flow velocity vector fields, the volume inflow rates into the aneurysm, and the wall shear stress (WSS) and wall pressure at the aneurysm dome. Some comparisons of computed results with the present FV and PTV results and with the data available from the literature are also made. The maximum flow velocity inside the aneurysm ostium and the WSS in the dome region at the peak flow can, respectively, be suppressed to less than 5% of the parent vessel bulk velocity (or 20% of the unstented case) and 8% of the unstented case if the stent porosity is smaller than 40% (about the porosity of the two-layer stents). In general, the three-layer stents seem not as effective as the two-layer stents in reducing the magnitude of aneurysm inflow rate and WSS.

A coupled finite volume method for the computational modelling of mould filling in very complex geometries

Simulation of shape casting processes involves the simultaneous capture of the rapidly developing metal–air free surface and the liquid–solid moving boundary in the metal phase. Although a wide range of techniques have been developed to address this problem, there are occasions where existing techniques are limited, especially in very complex geometries where the mesh quality may inevitably be relatively poor at some key locations. Cell-centred finite volume methods, typical of most commercial CFD tools, are computationally efficient, but can lead to convergence problems on meshes that feature cells with highly non-orthogonal shapes. The control volume–finite element method uses a vertex-based approach and handles distorted meshes with relative ease, but is computationally expensive. A combined vertex-based–cell-centre technique, described in this paper, solves the flow field at the cell or element vertex, and all other variables are solved at the cell centre (as in conventional CFD tools). This flow solver strategy is much more tolerant of meshes with poor quality elements, allowing solutions on distorted meshes where purely cell-centred solutions procedures fail. The combined method enables the solution of simultaneous free surface flows coupled with solidification in very complex meshes, thus enabling the capture of such behavior in arbitrarily complex geometrical structures. It is relatively straightforward to embed within generic CC based CFD tools allowing it to be employed in a wide variety of processing applications.