Cell-centered Finite Volume Method Research Articles

Numerical Investigations of the High-Lift Configuration with MFlow Solver

Numerical investigations of the DLR F11 high-lift configuration from the 2nd AIAA High-Lift Prediction Workshop are performed with the in-house solver MFlow. The solver is based on a cell-centered finite-volume method and is capable of handling various element types. Hexahedral grids and hybrid grids are used in the simulations. Grid-convergence properties, polar line, and pressure distributions are analyzed and compared with experimental data. A nearly linear convergence property is achieved with grid refinement for configuration 2, which is the most simplified configuration of the experiment model. When including slat and flap brackets (configuration 4), results are improved compared to the simplified configuration. However, delayed stall and larger maximum lift coefficient compared with experiment are found for high-Reynolds-number computations. The rotation correction of Spalart–Allmaras model does not lead to obvious improvement at high angles of attack. The effects of Reynolds number are qualitatively captured by the simulations. The inclusion of pressure-tube bundles causes loss of lift near stall and has little influence on predictions at lower angles of attack. The solver shows good agreement with experiment at lower angles of attack, but more attention is needed at angles of attack near and beyond stall.

A mass-conservative finite volume predictor–corrector solution of the 1D Richards’ equation

Summary Numerical solution of the Richards’ equation (RE) in variably saturated soils continues to be a challenge due to its highly non-linear behavior. This is particularly true as soils approach saturation and the behavior of the fundamental partial differential equation changes from elliptic to parabolic. In this paper, a finite volume predictor–corrector method with adaptive time-stepping was developed to solve the 1D vertical RE. The numerical method was mass-conservative and non-iterative. In the predictor step, the pressure head-based form of the RE was solved using the cell-centered finite volume method and the pressure head was updated. In the corrector step, the soil water content was calculated by solving the mixed form RE. Five different schemes to evaluate the inter-cell hydraulic conductivity were investigated. The robustness and accuracy of the numerical model were demonstrated through simulation of experimental tests, including free drainage, field infiltration into wet and dry soils, and laboratory infiltration with falling water table. Numerical results were compared against laboratory measurements, simulation results from the Hydrus-1D program, or analytical solution when available. Results showed that the developed scheme is robust and accurate in simulating variably saturated flows with various boundary conditions. The arithmetic mean and Szymkiewicz’s mean of inter-cell hydraulic conductivity performed better than other methods especially in the case of infiltration into very dry soil.

A higher-order macroscopic model for bi-direction pedestrian flow

In this paper, a higher-order macroscopic model for bi-direction pedestrian flow is presented to investigate macroscopic crowd patterns of bi-direction pedestrian movement. For each group, the macroscopic model consists of two-dimensional Euler equations with relaxation and an Eikonal-type equation. The magnitude of the desired velocity is described by an empirical relation between density and speed, while its direction is chosen to minimize the total instantaneous walking cost which satisfies a reactive user equilibrium assignment pattern. The dynamic model is numerically solved by a splitting technique and a cell-centered finite volume method on an orthogonal grid. Typical numerical examples of bi-direction pedestrian flow walking in a channel are designed to validate the rationality of the dynamic model and the effectiveness of the numerical algorithm.

Comparison between High Order Schemes Related Convection Diffusion of Navier-stokes Equations

In this paper, the cell-centered finite volume method with unstructured collocated meshes is applied to compare the accuracy of four schemes, the DC-CD, DC-SO, DCQ-QUICK and SGSD schemes are implemented for discrete convection. According to compare the numerical solutions with the analytical solutions, the impact of different formats for discrete numerical model calculation results are discussed. The results show the DCQ-QUICK scheme has the robust, accuracy and excellent convergence. Moreover, the DCQ-QUICK scheme is effective and trustable for solving complex regional problems.

Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity

We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.

A nonconforming a posteriori estimator for the coupling of cell-centered finite volume and boundary element methods

The coupling of the cell-centered finite volume method and the boundary element method is an interesting approach to solve elliptic problems on an unbounded domain, where local flux conservation is important. Based on the piecewise constant interior finite volume solution we define a Morley-type interpolant built on a nonconforming finite element. Together with the Cauchy data of the exterior boundary element solution this allows us to define a residual-based a posteriori error estimator. With respect to an energy norm we prove reliability and efficiency of this estimator and use its local contributions to steer an adaptive mesh-refining algorithm. In two examples we illustrate the effectiveness of the new adaptive coupling method and compare it with the coupling approach with a conforming Morley interpolant.

Robust a Posteriori Error Estimates for Conforming Discretizations of Diffusion Problems with Discontinuous Coefficients on Anisotropic Meshes

In this paper, we study a posteriori estimates for different numerical methods of diffusion problems with discontinuous coefficients on anisotropic meshes, in particular, which can be applied to vertex-centered and cell-centered finite volume, finite difference and piecewise linear finite element methods. Based on the stretching ratios of the mesh elements, we improve a posteriori estimates developed by Vohralik (J Sci Comput 46:397---438, 2011), which are reliable and efficient on isotropic meshes but fail on anisotropic ones (see the numerical results of the paper). Without the assumption that the meshes are shape-regular, the resulting mesh-dependent error estimators are shown to be reliable and efficient with respect to 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, as long as the anisotropic mesh sufficiently reflects the anisotropy of the solution. In other words, they are equivalent to the estimates of Vohralik in the case of isotropic meshes and proved to be robust on anisotropic meshes as well. Based on $$\mathbf{H}(\mathrm {div})$$H(div)-conforming, locally conservative flux reconstruction, we suggest two different constructions of the equilibrated flux with the anisotropy of mesh, which is essential to the robustness of our estimates on anisotropic meshes. Numerical experiments in 2D confirm that our estimates are reliable and efficient on anisotropic meshes.

Parallel performance modeling of irregular applications in cell-centered finite volume methods over unstructured tetrahedral meshes

Finite volume methods are widely used numerical strategies for solving partial differential equations. This paper aims at obtaining a quantitative understanding of the achievable performance of the cell-centered finite volume method on 3D unstructured tetrahedral meshes, using traditional multicore CPUs as well as modern GPUs. By using an optimized implementation and a synthetic connectivity matrix that exhibits a perfect structure of equal-sized blocks lying on the main diagonal, we can closely relate the achievable computing performance to the size of these diagonal blocks. Moreover, we have derived a theoretical model for identifying characteristic levels of the attainable performance as a function of hardware parameters, based on which a realistic upper limit of the performance can be predicted accurately. For real-world tetrahedral meshes, the key to high performance lies in a reordering of the tetrahedra, such that the resulting connectivity matrix resembles a block diagonal form where the optimal size of the blocks depends on the hardware. Numerical experiments confirm that the achieved performance is close to the practically attainable maximum and it reaches 75% of the theoretical upper limit, independent of the actual tetrahedral mesh considered. From this, we develop a general model capable of identifying bottleneck performance of a system’s memory hierarchy in irregular applications.

Numerical analysis of unsteady cavitating flow in an axial inducer

This work presents the results of numerical simulation of unsteady cavitating flow through a two–bladed axial inducer. First, the analysis was carried out in a blade cascade, this two–dimensional simplified model, obtained from the studied axial inducer, was used as a test case. Later, the numerical simulations were extended to the original three-dimensional inducer. All numerical calculations were realized in cavitating flow regime. Initially, the results were obtained in steady state, and then in unsteady state.The main purpose of this study is to explore the local cavitation instabilities, such as alternate blade cavitation and rotating blade cavitation, which can appear in this type of devices when they work under certain operating conditions.The numerical results show that the fluid flow in the axial inducer is altered by the emergence of the cavitation. These vapor regions are formed, firstly near to the leading edge of each blade. The behavior of the cavitation depends on the operating conditions of the inducer, mainly by the flow rate and the suction pressure.The numerical simulation was performed using a commercial code based on a cell–centered 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 vapor phase and a mixture phase, with mass transfer due to the cavitation appearing as a source and a sink term in the vapor mass fraction equation. The mass transfer rate is derived from a simplified Rayleigh–Plesset model for bubble dynamics.

Development of a Novel Fully Coupled Solver in OpenFOAM: Steady-State Incompressible Turbulent Flows

In this work a block coupled algorithm for the solution of three-dimensional incompressible turbulent flows is presented. A cell-centered finite-volume method for unstructured grids is employed. The interequation coupling of the incompressible Navier-Stokes equations is obtained using a SIMPLE-type algorithm with a Rhie-Chow interpolation technique. Due to the simultaneous solution of momentum and continuity equations, implicit block coupling of pressure and velocity variables leads to faster convergence compared to classical, loosely coupled, segregated algorithms of the SIMPLE family of algorithms. This gain in convergence speed is accompanied by an improvement in numerical robustness. Additionally, a two-equation eddy viscosity turbulence model is solved in a segregated fashion. The substnatially improved performance of the block coupled approach compared to the segregated approach is demonstrated in a set of test cases. It is shown that the scalability of the coupled solution algorithm with increasing numbers of cells is nearly linear. To achieve this scalability, an algebraic multigrid solver for block coupled systems of equations has been implemented and used as linear solver for the system of block equations. The presented algorithm has been entirely embedded into the leading open-source computational fluid dynamics (CFD) library OpenFOAM.

Finite volume hydromechanical simulation in porous media.

Cell-centered finite volume methods are prevailing in numerical simulation of flow in porous media. However, due to the lack of cell-centered finite volume methods for mechanics, coupled flow and deformation is usually treated either by coupled finite-volume-finite element discretizations, or within a finite element setting. The former approach is unfavorable as it introduces two separate grid structures, while the latter approach loses the advantages of finite volume methods for the flow equation. Recently, we proposed a cell-centered finite volume method for elasticity. Herein, we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media. We detail in particular the issue of coupling terms, and show how this is naturally handled. Furthermore, we observe how the cell-centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions. We support the discussion with a set of numerical examples: the convergence properties of the coupled scheme are first investigated; second, we illustrate the practical applicability of the method both for fractured and heterogeneous media.

Nonlinear and dispersive free surface waves propagating over fluids with weak vertical and horizontal density variation

AbstractWe consider the change in fluid density in a depth-integrated long-wave model. By allowing horizontal and vertical variation of fluid density, a depth-integrated model for long gravity waves over a variable-density fluid has been developed, where density change effects are included as correction terms. In particular, a two-layer fluid system is chosen to represent vertical density variations, where interfacial wave effects on the free surface are accounted for through direct inclusion of the velocity component of the interfacial wave. For the numerical implementation of the model, a finite-volume scheme coupled with an approximate Riemann solver is adopted for leading-order terms while cell-centred finite-volume methods are utilized for others. Numerical tests in which the density field is configured to vary either horizontally or vertically have been performed to verify the model. For horizontal variation of fluid density, a pneumatic breakwater system is simulated and fair agreement is observed between computed and measured data, indicating that the current induced by the upward bubble flux is responsible for wave attenuation to some degree. To investigate the effects of internal motion on the free surface, a two-layer fluid system with monochromatic internal wave motion is tested numerically. Simulated results agree well with the measured and analytical data. Lastly, nonlinear interactions between external- and internal-mode surface waves are studied numerically and analytically, and the model is shown to have nonlinear accuracy limitations similar to existing Boussinesq-type models.

Large eddy simulation of a randomly stacked nuclear pebble bed

This paper reports large eddy simulation results for a randomly stacked bed of spherical pebbles, using a second-order accurate, cell-centered finite volume method on an unstructured polyhedral mesh. The selected flow configuration represents the core of a high temperature reactor, in which nuclear fuel is embedded in the pebbles. The geometrical arrangement consists of approximately 30 pebbles, which are randomly stacked and in contact with each other. Extensive analyses of flow and thermal fields are performed to derive valuable insights on the flow characteristics. The predicted flow-field is fairly complex and exhibits highly unsteady and three-dimensional turbulent behavior with strong rotational and cross flow regions. The flow, while moving over the pebbles shows attenuation and enhancement in the turbulence levels, and eventually yields to flow separation and its subsequent reattachment. Consequently, a wide range of temperatures over the pebbles is observed; this includes the appearance of hot-spots especially around contact areas. The occurrence of high- and low-velocity streak regions, and the corresponding temperature fields are examined and quantified. The reported analyses can be used for validation of low-order turbulence modeling applications to complex pebble flow configurations.

Numerical solution of unsteady generalized Newtonian and Oldroyd-B fluids flow by dual time-stepping method

This work deals with the numerical solution of viscous and viscoelastic fluids flow. The governing system of equations is based on the system of balance laws for mass and momentum for incompressible laminar fluids. Different models for the stress tensor are considered. For viscous fluids flow Newtonian model is used. For the describing of the behaviour of the mixture of viscous and viscoelastic fluids Oldroyd-B model is used. Numerical solution of the described models is based on cell-centered finite volume method in conjunction with artificial compressibility method. For time integration an explicit multistage Runge-Kutta scheme is used. In the case of unsteady computation dual-time stepping method is considered. The principle of dual-time stepping method is following. The artificial time is introduced and the artificial compressibility method in the artificial time is applied.

Vertex-based finite volume simulation of Liesegang patterns on structureless meshes.

A computational method is suggested for the simulation of Liesegang patterns in two dimensions on structureless meshes. The method is based on a model that incorporates dynamical equations for the nucleation and growth of solid particles of different sizes into reaction-diffusion equations. We find the model cannot be numerically solved with Galerkin-based finite element methods and cell-centered finite volume methods. Instead, the vertex-based finite volume method is used to correctly reproduce the Liesegang pattern on structureless meshes. The numerical solution is then compared with specially designed experiments on Liesegang patterns in various geometries, and it is shown to be in good agreement.

Numerical Simulation of Unsteady Aerodynamic Loads over an Aircraft

The unsteady aerodynamic loads are the basic of the aeroelastic. This paper focuses on the computation of the unsteady aerodynamic loads for forced periodic motions under different Mach numbers. The flow is modeled using the Euler equations and an unsteady time-domain solver is used for the computation of aerodynamic loads for forced periodic motions. The Euler equations are discretized on curvilinear multi-block body conforming girds using a cell-centred finite volume method. The implicit dual-time method proposed by Jameson is used for time-accurate calculations. Rigid body motions were treated by moving the mesh rigidly in response to the applied sinusoidal motion. For an aircraft model, a validation of the unsteady aerodynamics loads is first considered. Furthermore, a study for understanding the influence of different Mach number was conducted. A reverse of the trend of hysteretic loops can be observed with the increasing of the Mach number.

The application of eulerian two-phase flow method in airfoil ice accretion

A numerical simulation method based on Eulerian Two-Phase theory was applied on airfoil under a variety of icing weather conditions. The flow field is calculated by solving the 2D N-S equations with cell-centered finite volume method. The Rime ice, Glaze ice and mixed ice were predicted on NACA0012 airfoil at 4 degree angle of attack. The simulated results agree very well with the refered experiment data, which indicates that the method is effective and correct.At last the aerodynamic characteristics of the iced airfoil was analyzed, compare different ice shape on the lift and drag coefficient, the results show that the Glaze ice has the most impact on aerodynamic characteristics.

Modeling and Simulation of Compressible Three-Phase Flows in an Oil Reservoir: Case Study of Tsimiroro Madagascar

Oil extraction represents an important investment and the control of a rational exploitation of a field means mastering various scientific techniques including the understanding of the dynamics of fluids in place. This paper presents a theoretical investigation of the dynamic behavior of an oil reservoir during its exploitation. More exactly, the mining process consists in introducing a miscible gas into the oil phase of the field by means of four injection wells which are placed on four corners of the reservoir while the production well is situated in the middle of this one. So, a mathematical model of multiphase multi-component flows in porous media was presented and the cell-centered finite volume method was used as discretization scheme of the considered model equations. For the simulation on Matlab, the case of the oil-field of Tsimiroro Madagascar was studied. It ensues from the analysis of the contour representation of respective saturations of oil, gas and water phases that the conservation law of pore volume is well respected. Besides, the more one moves away from the injection wells towards the production well; the lower is the pressure value. However, an increase of this model variable value was noticed during production period. Furthermore, a significant accumulated flow of oil was observed at the level of the production well, whereas the aqueous and gaseous phases are there present in weak accumulated flow. The considered model so allows the prediction of the dynamic behavior of the studied reservoir and highlights the achievement of the exploitation process aim.

Development of a Cell-Centered Godunov-Type Finite Volume Model for Shallow Water Flow Based on Unstructured Mesh

Based on the Godunov-type cell-centered finite volume method, this paper presents a two-dimensional well-balanced shallow water model for simulating flows over arbitrary topography with wetting and drying. The central upwind scheme is used for the computation of mass and momentum fluxes on interface. The novel aspect of the present model is a robust and accurate nonnegative water depth reconstruction method which is implemented in the unstructured mesh to achieve second-order accuracy in space and to track the moving wet/dry fronts of the flow over irregular terrain. By defining the bed elevation and primary flow variables at the cell center in the nonstaggered grid system, all computational cells are either fully wet or dry to avoid the problem of being partially wetted. The developed model is capable of being well balanced and preserving the computed water depth to be nonnegative under a certain CFL restriction, which makes it robust and stable. The present model is validated against three benchmark tests and two laboratory dam-break cases. Finally, the good agreement between the numerical results by the established model and measured data of the Malpasset dam break event on a 1/400 scale physical model demonstrates the capability of the model for the real-life applications.

Local Mass-Corrections for Continuous Pressure Approximations of Incompressible Flow

In this work, we discuss a family of finite element discretizations for the incompressible Stokes problem using continuous pressure approximations on simplicial meshes. We show that after a simple and cheap correction, the mass-fluxes obtained by the considered schemes preserve local conservation on dual cells without reducing the convergence order. This allows the direct coupling to vertex-centered finite volume discretizations of transport equations. Further, we can postprocess the mass fluxes independently for each dual box to obtain an elementwise conservative velocity approximation of optimal order that can be used in cell-centered finite volume or discontinuous Galerkin schemes. Numerical examples for stable and stabilized methods are given to support our theoretical findings. Moreover, we demonstrate the coupling to vertex- and cell-centered finite volume methods for advective transport.