Conservative Finite Volume Research Articles

Flow Characteristics in Mixers Agitated by Helical Ribbon Blade Impeller

The main concern of this study is to investigate the flow mixing generated by helical ribbon blade impellers and to show that with the help of CFD the performance of the mixing system can be significantly improved by optimizing the geometric configuration of the impeller. To fulfill this objective, a numerical model is developed to solve the Navier-Stokes equations for the flow field. However, difficulties arise due to the rotation of the impeller in the vessel. In order to ease the problem, the velocity field is assumed to be in a quasi-steady state and the multiframe of reference is adopted to tackle the rotation of the impeller. For discretization the fully conservative finite volume method, together with unstructured grid technology, is incorporated. It is shown that the flow in the mixer can be regarded as a flow in an open channel with a wall moving at an angle with respect to the channel. The influences of the blade pitch, the blade width, and the clearance gap between the blade and the surrounding wall are examined. The mechanism to cause these effects is delineated in detail. It is demonstrated that after optimization of the blade geometry, the circulating flow rate induced by the impeller is largely increased, leading to significant reduction in mixing time. In addition, the power demand is reduced. It is also evidenced that by enlarging the clearance, it is difficult for the fluid in this region to be mixed.

A convergence of a MFE-FV method for immiscible compressible flow in heterogeneous porous media

Abstract This paper deals with the development and analysis of a numerical method for a coupled system describing immiscible compressible two-phase flow through heterogeneous porous media. The system is modelled in a fractional flow formulation which consists of a parabolic equation (the global pressure equation) coupled with a nonlinear degenerated diffusion-convection one (the saturation equation). A mixed finite element (MFE) method is used to discretize the pressure equation and is combined with a conservative finite volume (FV) method on unstructured grids for the saturation equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L ∞ and BV estimates under an appropriate CFL condition. Then we prove the convergence of the approximate solution to a weak solution of the coupled system. Numerical results for water-gas flow through engineered and geological barriers for a geological repository of radioactive waste are presented to illustrate the performance of the method in two space dimensions.

Thermal analysis of a cylindrical lithium-ion battery

This work investigates the heat generation characteristics of a cylindrical lithium-ion battery. The battery consists of the graphite, LiPF 6 of the propylene carbonate/ethylene carbonate/dimethyl carbonate (PC/EC/DMC) solution, and spinal as anode, electrolyte and cathode, respectively. The coupled electrochemical–thermal model is developed with full consideration of electrolyte transport properties as functions of temperature and Li ion concentration. A truly conservative finite volume numerical method is employed for the spatial discretization of the model equations. Three types of heat generation sources including the ohmic heat, the active polarization heat and the reaction heat are quantitatively analyzed for the battery discharge process. The ohmic heat is found to be the largest contribution with around 54% in the total heat generation. About 30% of the total heat generation in average is ascribed to the electrochemical reaction. The active polarization contributes the least comparing to the ohmic heat and reactions heat. The results also show that the Li ion concentration and its gradient in electrolyte are the main factors giving the effect on the heat generations of active polarization and electrolyte electric resistance. The raised temperature in the battery discharge is positive related with the thickness of both separator and electrodes.

CFD simulation of laser enhanced modified chemical vapor deposition process

The modified chemical vapor deposition (MCVD) process is one of the most widely used processes to manufacture the optical fiber preforms. There have been several experimental and numerical studies to establish that thermophoresis is the dominant mechanism for the transport of silica particles to the preform wall. There also exists several modification of this process to increase the deposition efficiency in the MCVD process. In this work we have carried out computational fluid dynamics simulation of the coupled equations of mass, momentum, energy and species transport in the laser enhanced thermophoretic deposition process using the conservative finite volume scheme. The effect of laser heating on different parameters such as thermophoresis, conversion rate and deposition efficiency is examined in this study. The presence of laser heating alters the velocity and temperature profile that leads to increase in the conversion and deposition efficiency due to high thermophoresis. The results of numerical simulations are in support of experimental and analytical studies. Our numerical simulations using the conservative finite volume scheme show that deposition efficiency increases with increasing power of the laser.

A conservative and monotone mixed-hybridized finite element approximation of transport problems in heterogeneous domains

In this article, we discuss the numerical approximation of transport phenomena occurring at material interfaces between physical subdomains with heterogenous properties. The model in each subdomain consists of a partial differential equation with diffusive, convective and reactive terms, the coupling between each subdomain being realized through an interface transmission condition of Robin type. The numerical approximation of the problem in the two-dimensional case is carried out through a dual mixed-hybridized finite element method with numerical quadrature of the mass flux matrix. The resulting method is a conservative finite volume scheme over triangular grids, for which a discrete maximum principle is proved under the assumption that the mesh is of Delaunay type in the interior of the domain and of weakly acute type along the domain external boundary and internal interface. The stability, accuracy and robustness of the proposed method are validated on several numerical examples motivated by applications in biology, electrophysiology and neuroelectronics.

Robust multigrid preconditioners for cell-centered finite volume discretization of the high-contrast diffusion equation

We study a conservative 5-point cell-centered finite volume discretization of the high-contrast diffusion equation. We aim to construct preconditioners that are robust with respect to the magnitude of the coefficient contrast and the mesh size simultaneously. For that, we prove and numerically demonstrate the robustness of the preconditioner proposed by Aksoylu et al. (Comput Vis Sci 11:319–331, 2008) by extending the devised singular perturbation analysis from linear finite element discretization to the above discretization. The singular perturbation analysis is more involved than that of finite element case because all the subblocks in the discretization matrix depend on the diffusion coefficient. However, as the diffusion coefficient approaches infinity, that dependence is eliminated. This allows the same preconditioner to be utilized due to similar limiting behaviours of the submatrices; leading to a narrowing family of preconditioners that can be used for different discretizations. Therefore, we have accomplished a desirable preconditioner design goal. We compare our numerical results to standard cell-centered multigrid implementations and observe that performance of our preconditioner is independent of the utilized smoothers and prolongation operators. As a side result, we also prove a fundamental qualitative property of solution of the high-contrast diffusion equation. Namely, the solution over the highly-diffusive island becomes constant asymptotically. Integration of this qualitative understanding of the underlying PDE to our preconditioner is the main reason behind its superior performance. Diagonal scaling is probably the most basic preconditioner for high-contrast coefficients. Extending the matrix entry based spectral analysis introduced by Graham and Hagger, we rigorously show that the number of small eigenvalues of the diagonally scaled matrix depends on the number of isolated islands comprising the highly-diffusive region. This indicates that diagonal scaling creates a significant clustering of the spectrum, a favorable property for faster convergence of Krylov subspace solvers.

Finite-Volume Formulation and Solution of the P3 Equations of Radiative Transfer on Unstructured Meshes

Abstract The method of spherical harmonics (or PN) is a popular method for approximate solution of the radiative transfer equation (RTE) in participating media. A rigorous conservative finite-volume (FV) procedure is presented for discretization of the P3 equations of radiative transfer in two-dimensional geometry—a set of four coupled, second-order partial differential equations. The FV procedure presented here is applicable to any arbitrary unstructured mesh topology. The resulting coupled set of discrete algebraic equations are solved implicitly using a coupled solver that involves decomposition of the computational domain into groups of geometrically contiguous cells using the binary spatial partitioning algorithm, followed by fully implicit coupled solution within each cell group using a preconditioned generalized minimum residual solver. The RTE solver is first verified by comparing predicted results with published Monte Carlo (MC) results for two benchmark problems. For completeness, results using the P1 approximation are also presented. As expected, results agree well with MC results for large/intermediate optical thicknesses, and the discrepancy between MC and P3 results increase as the optical thickness is decreased. The P3 approximation is found to be more accurate than the P1 approximation for optically thick cases. Finally, the new RTE solver is coupled to a reacting flow code and demonstrated for a laminar flame calculation using an unstructured mesh. It is found that the solution of the four P3 equations requires 14.5% additional CPU time, while the solution of the single P1 equation requires 9.3% additional CPU time over the case without radiation.

Flux-blending schemes for interface capture in two-fluid flows

This paper is aimed at developing a robust algorithm (FBICS) to capture the interface between two immiscible fluids without the need of interface reconstruction. The advection equation of the volume fraction is solved using the fully conservative finite volume method. Determination of the convective flux through each cell face is based on blending of high resolution schemes and compressive schemes to preserve the sharpness and boundedness of the interface. The flux-blending practice is fulfilled with the use of flux limiters. Test on simple advection flow problems indicates that the well-known CICSAM and HRIC schemes lose accuracy as the Courant number increases. In contrast, the present method maintains high-accuracy performance for Courant numbers up to one. The capability of the method to cope with the complicated dynamics of free surface flows is demonstrated via calculation of the collapsing flow of a water column with an obstacle.

Use of Characteristic-Based Flux Limiters in a Pressure-Based Unstructured-Grid Algorithm Incorporating High-Resolution Schemes

A pressure-based procedure to solve problems ranging from incompressible to highly compressible flows is described. The method adopts the fully conservative finite-volume approach, for which the meshes can be of any topology. To handle the sharp change of the gradient in the regions near the shock or the solid wall, the convective flux is limited using high-resolution schemes such as the total variation diminishing (TVD) or the normalized variable diagram (NVD) scheme. The flux limiters are determined from the characteristic variables instead of the commonly used primitive or conservative variables. To enhance solution accuracy, the gradient is calculated using a linear reconstruction approach. The method is assessed and validated via testing on a number of flow problems, including inviscid flows through a convergent-divergent nozzle, inviscid and viscous flows past an airfoil, and viscous flows through a double-throat nozzle.

A finite volume local defect correction method for solving the transport equation

The local defect correction (LDC) method is applied in combination with standard finite volume discretizations to solve the advection–diffusion equation for a passive tracer. The solution is computed on a composite grid, i.e. a union of a global coarse grid and local fine grids. For the test a dipole colliding with a no-slip wall is used to provide an actively changing velocity field. The LDC method is tested for the problem of localized patch of tracer material that is transported by the provided velocity field. The LDC algorithm can be formulated to conserve the total amount of tracer material. However, if the local fine grids are moved to adaptively follow the behaviour of the solution, a loss or gain in the total amount of tracer material is produced. This deficit in tracer material is created when the solution is interpolated to obtain data for the moved fine grid. The data obtained by the interpolation scheme in the new refined region can be adapted in such a way that the surplus or deficit is spread over the new grid points and conservation of tracer material is satisfied. Finally, the results of the conservative finite volume LDC method are compared and validated with results from a spectral method.

Numerical modeling of the flow and transport of radionuclides in heterogeneous porous media

This paper is concerned with numerical methods for the modeling of flow and transport of contaminant in porous media. The numerical methods feature the mixed finite element method over triangles as a solver to the Darcy flow equation and a conservative finite volume scheme for the concentration equation. The convective term is approximated with a Godunov scheme over the dual finite volume mesh, whereas the diffusion–dispersion term is discretized by piecewise linear conforming triangular finite elements. It is shown that the scheme satisfies a discrete maximum principle. Numerical examples demonstrate the effectiveness of the methodology for a coupled system that includes an elliptic equation and a diffusion–convection–reaction equation arising when modeling flow and transport in heterogeneous porous media. The proposed scheme is robust, conservative, efficient, and stable, as confirmed by numerical simulations.

Hall magnetohydrodynamics on block-adaptive grids

We present a conservative second order accurate finite volume discretization of the magnetohydrodynamics equations including the Hall term. The scheme is generalized to three-dimensional block-adaptive grids with Cartesian or generalized coordinates. The second order accurate discretization of the Hall term at grid resolution changes is described in detail. Both explicit and implicit time integration schemes are developed. The stability of the explicit time integration is ensured by including the whistler wave speed for the shortest discrete wave length into the numerical dissipation, but then second order accuracy requires the use of symmetric limiters in the total variation diminishing scheme. The implicit scheme employs a Newton–Krylov–Schwarz type approach, and can achieve significantly better efficiency than the explicit scheme with an appropriate preconditioner. The second order accuracy of the scheme is verified by numerical tests. The parallel scaling and robustness are demonstrated by three-dimensional simulations of planetary magnetospheres.

Mixing and evaporation of liquid droplets injected into an air stream flowing at all speeds

This paper deals with the formulation, implementation, and testing of three numerical techniques based on (i) a full multiphase approach, (ii) a multisize-group (MUSIG) approach, and (iii) a heterogeneous MUSIG (H-MUSIG) approach for the prediction of mixing and evaporation of liquid droplets injected into a stream of air. The numerical procedures are formulated following an Eulerian approach, within a pressure-based fully conservative finite volume method equally applicable in the subsonic, transonic, and supersonic regimes, for the discrete and continuous phases. The k-ε two-equation turbulence model is used to account for the droplet and gas turbulence with modifications to account for compressibility at high speeds. The performances of the various methods are compared by solving for two configurations involving streamwise and cross-stream sprayings into subsonic and supersonic streams. Results, which are displayed in the form of droplet velocity vectors, contour plots, and axial profiles, indicate that solutions obtained by the various techniques exhibit a similar behavior. Differences in values are relatively small with the largest being associated with droplet volume fractions and vapor mass fraction in the gas phase. This is attributed to the fact that with MUSIG and H-MUSIG, no droplet diameter equation is solved and the diameter of the various droplet phases is held constant, as opposed to the full multiphase approach.

Numerical investigation of three-dimensional cloud cavitation with special emphasis on collapse induced shock dynamics

The aim of the present investigation is to model and analyze compressible three-dimensional (3D) cavitating liquid flows with special emphasis on the detection of shock formation and propagation. We recently developed the conservative finite volume method CATUM (Cavitation Technische Universität München), which enables us to simulate unsteady 3D liquid flows with phase transition at all Mach numbers. The compressible formulation of the governing equations together with the thermodynamic closure relations are solved by a modified Riemann approach by using time steps down to nanoseconds. This high temporal resolution is necessary to resolve the wave dynamics that leads to acoustic cavitation as well as to detect regions of instantaneous high pressure loads. The proposed two-phase model based on the integral average properties of thermodynamic quantities is first validated against the solution of the Rayleigh–Plesset equation for the collapse of a single bubble. The computational fluid dynamics tool CATUM is then applied to the numerical simulation of the highly unsteady two-phase flow around a 3D twisted hydrofoil. This specific hydrofoil allows a detailed study of sheet and cloud cavitation structures related to 3D shock dynamics emerging from collapsing vapor regions. The time dependent development of vapor clouds, their shedding mechanism, and the resulting unsteady variation of lift and drag are discussed in detail. We identify instantaneous local pressure peaks of the order of 100bar, which are thought to be responsible for the erosive damage of the surface of the hydrofoil.

A conservative characteristic finite volume element method for solution of the advection–diffusion equation

We develop a modified characteristic finite volume element method for two-dimensional advection–diffusion problem in a convex spacial domain, using quasi-uniform triangular discretization and piecewise linear element. This scheme conserves mass globally, and locally on the Lagrangian space–time control volume. We give the optimal order H 1-norm and optimal L 2-norm error estimates. Finally we present some numerical examples.

Formulation of Kinetic Energy Preserving Conservative Schemes for Gas Dynamics and Direct Numerical Simulation of One-Dimensional Viscous Compressible Flow in a Shock Tube Using Entropy and Kinetic Energy Preserving Schemes

This paper follows up on the author's recent paper Construction of Discretely Conservative Finite Volume Schemes that also Globally Conserve Energy or Enthalpy. In the case of the gas dynamics equations the previous formulation leads to an entropy preserving (EP) scheme. It is shown in the present paper that it is also possible to construct the flux of a conservative finite volume scheme to produce a kinetic energy preserving (KEP) scheme which exactly satisfies the global conservation law for kinetic energy. A proof is presented for three dimensional discretization on arbitrary grids. Both the EP and KEP schemes have been applied to the direct numerical simulation of one-dimensional viscous flow in a shock tube. The computations verify that both schemes can be used to simulate flows with shock waves and contact discontinuities without the introduction of any artificial diffusion. The KEP scheme performed better in the tests.

The Construction of Discretely Conservative Finite Volume Schemes that Also Globally Conserve Energy or Entropy

This work revisits an idea that dates back to the early days of scientific computing, the energy method for stability analysis. It is shown that if the scalar non-linear conservation law $$\frac{\partial u}{\partial t}+\frac{\partial}{\partial x}f(u)=0$$ is approximated by the semi-discrete conservative scheme $$\frac{du_{j}}{dt}+\frac{1}{\Delta x}\left(f_{j+\frac{1}{2}}-f_{j-\frac{1}{2}}\right)=0$$ then the energy of the discrete solution evolves at exactly the same rate as the energy of the true solution, provided that the numerical flux is evaluated by the formula $$f_{j+\frac{1}{2}}=\int_{0}^{1}f(\hat{u})d\theta,$$ where $$\hat{u}(\theta)=u_{j}+\theta(u_{j+1}-u_{j}).$$ With careful treatment of the boundary conditions, this provides a path to the construction of non-dissipative stable discretizations of the governing equations. If shock waves appear in the solution, the discretization must be augmented by appropriate shock operators to account for the dissipation of energy by the shock waves. These results are extended to systems of conservation laws, including the equations of incompressible flow, and gas dynamics. In the case of viscous flow, it is also shown that shock waves can be fully resolved by non-dissipative discretizations of this type with a fine enough mesh, such that the cell Reynolds number ≤2.

A positive and conservative second order finite volume scheme applied to a phosphor cycle in canals with sediment

AbstractThis paper is devoted to the description of a higher order finite volume scheme for the 2D shallow water equation with a specific focus on a complex phosphor cycle describing processes in lakes and rivers. The scheme guarantees the unconditional positivity and mass conservation of respectively modelled quantities, which are crucial properties for meaningful computed results in biochemical contexts. It works on secondary unstructured meshes which enables the application to arbitrary complex geometries. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convergence analysis of an approximation to miscible fluid flows in porous media by combining mixed finite element and finite volume methods

AbstractThis article deals with development and analysis of a numerical method for a coupled system describing miscible displacement of one incompressible fluid by another through heterogeneous porous media. A mixed finite element (MFE) method is employed to discretize the Darcy flow equation combined with a conservative finite volume (FV) method on unstructured grids for the concentration equation. It is shown that the FV scheme satisfies a discrete maximum principle. We derive L∞ and BV estimates under an appropriate CFL condition. Then we prove convergence of the approximate solutions to a weak solution of the coupled system. Numerical results are presented to see the performance of the method in two space dimensions. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008

Multiscale modeling of turbulent combustion and NOx emission in steam crackers

AbstractA 3D RANS simulation of lean premixed turbulent combustion in an industrial scale steam cracking furnace is performed. Turbulence is represented by a renormalization group (RNG) k − ε model, while combustion is modeled with a skeletal mechanism. The eddy dissipation concept (EDC) model is used to account for turbulence/combustion interaction. The turbulent combustion regime is identified as a “thin flame regime with pockets” or a “corrugated flamelet regime”. A weighted sum of gray gases model (WSGGM) is used to account for the gas radiative properties. The radiative transfer equation is solved using the P‐1 and discrete ordinates method in the conservative finite volume formulation. NOx calculations are performed as a postprocessing step, with the flow field, temperature, and species concentrations fixed. The formation of NO through thermal, prompt, and N2O intermediate mechanisms is considered. The in situ adaptive tabulation technique is used to decrease the computational time required to treat the combustion chemistry. The effect of radiation models on the predicted furnace wall, tube skin, and flue gas temperature profiles and heat fluxes toward the reactor tubes, as well as on the predicted species and NO concentration profiles and structure of the furnace flames under normal firing conditions, is discussed. © 2007 American Institute of Chemical Engineers AIChE J, 2007