Nite Volume Scheme Research Articles

A second order in space combination of methods verifying a maximum principle for the discretization of diffusion operators

We describe a second order in space combination of methods suppressing oscillations appearing for diffusion operator discretization with cell-centered nite volume schemes.

Numerical analysis for a three interacting species model with nonlocal and cross diffusion

In this paper, we consider a reaction-di usion system describing three interacting species in the food chain structure with nonlocal and cross di usion. We propose a semi implicit fi nite volume scheme for this system, we establish existence and uniqueness of the discrete solution, and it is also showed that the discrete solution generated by the given scheme converges to the corresponding weak solution for the model studied. The convergence proof is based on the use of the discrete Sobolev embedding inequalities with general boundary conditions and a space-time L1 compactness argument that mimics the compactness lemma due to S. N. Kruzhkov. Finally we give some numerical examples.

Finite volume scheme for isotropic Keller-Segel model with general scalar diffusive functions

This paper is devoted to the numerical analysis of a modied Keller-Segel model consisting of diusion and chemotaxis with volume lling eect. Firstly, a nite volume scheme is generalized to the case of a Keller-Segel model allowing heterogeneities and discontinuities in the diusion coecients. For that, we start with the derivation of the discrete problem and then we establish a convergence result of the discrete solution to a weak solution of the continuous model. Finally, numerical tests illustrate the behavior of the solutions of this generalized numerical scheme. R

Monotone Combined Finite Volume-Finite Element Scheme for Anisotropic Keller-Segel Model

The directed movement of cells and organisms in response to chemical gradients, Chemotaxis, has attracted signicant interest due to its critical role in a wide range of biological phenomena. The Keller-Segel model has provided a cornerstone for mathematical modeling of chemotaxis. We are interested by the numerical analysis of the degenerate Keller-Sege lmodel. A scheme recently developed in the nite volume framework treats the discretization of the model in homogeneous domains where the diffusion tensors are considered to be the identity matrix. However, standard nite volume scheme not permit to handle anisotropic diffusion on general,possibly nonconforming meshes. In the other hand, it is well-known that nite element discretization allows a very simple discretization of full diffusion tensors and does not impose any restrictions on the meshes but many numerical instabilities may arise in the convection-dominated case. A quite intuitive idea is hence to combine a nite element discretization of the diffusion term with a nite volume discretization of the other terms. Hence, we construct and study the convergence analysis of a combined scheme discretizing the system. This scheme ensures the validity of the discrete maximum principle under the classical condition that all transmissibilities coefficients are positive. Therefore, a nonlinear technique is presented, in the spirit of well-known methods as a correction to provide a monotone scheme for generaltensors.

Pattern Formation and Cross-Diffusion for a Chemotaxis Model

Chemotaxis is the feature movement of cell or an organism along a chemical concentration gradient. The mathematical analysis of chemotaxis modelss how a plenitude of spatial patterns such as the chemotaxis models applied to skin pigmentation patterns, that lead to aggregations of one type of pigment cells into a striped spatial pattern. The analysis of pattern formation can be traced to a seminal paper by Turing [1], who established that an action-diffusion system can generate stable nonuniform patterns in space if the components of the system interact with each other.Our motivation is the numerical simulations of the pattern formation for a volume-filling chemotaxis model. In [2], the effect of volume-filling is expressed through a nonlinear squeezing probability. We investigate pattern formation using Turing's principle and the standard argument used by Murray [3,4]. Next, we introduce an implicit nite volume scheme; it is presented on a general mesh satisfying the orthogonality condition [5,6].The originality of this scheme is the upstream approach to discretize thecross-diffusion term. Finally, we present some numerical results showing the spatial patterns for the chemotaxis model.

A SYMMETRIC FINITE VOLUME ELEMENT SCHEME ON TETRAHEDRON GRIDS

Abstract. We construct a symmetric nite volume element (SFVE)scheme for a self-adjoint elliptic problem on tetrahedron grids and provethat our new scheme has optimal convergent order for the solution andhas superconvergent order for the ux when grids are quasi-uniform andregular. The symmetry of our scheme is helpful to solve eciently thecorresponding discrete system. Numerical experiments are carried out tocon rm the theoretical results. 1. IntroductionDue to the local conservation property, the nite volume element methods[1, 4, 5, 6, 11, 13, 14, 23] have been greatly popular in many elds, such ascomputational uid dynamics, computational electromagnetic and petroleumengineering and so on.In many cases, the symmetry is the fundamental physical principle of reci-procity. Hence, it is signi cant and important to present a symmetric discretescheme. It is well known that the standard nite volume element (FVE) meth-ods [4, 6, 9, 14, 17] usually generate a linear system with asymmetric matrix.The asymmetry leads to the fact that many ecient iterative methods whichare suitable for solving the symmetric linear systems, such as the preconditionconjugate gradient (PCG) method, can’t be employed. Some SFVE schemes[16, 18, 21, 22, 24] essentially overcome the above defect for self-adjoint ellipticboundary-value and parabolic problems on triangular and quadrilateral grids.There are many nite volume schemes [1, 5, 19, 20, 26] constructed for solvingthree dimension problems. However, few scheme is symmetric so far. It hasmotivated us to propose a new symmetric scheme.

Adaptive multiresolution or adaptive mesh refinement? A case study for 2D Euler equations

We present adaptive multiresolution (MR) computations of the two-dimensional com- pressible Euler equations for a classical Riemann problem. The results are then compared with respect to accuracy and computational eciency, in terms of CPU time and memory requirements, with the corresponding nite volume scheme on a regular grid. For the same test case, we also perform compu- tations using adaptive mesh renement (AMR) imposing similar accuracy requirements. The results thus obtained are compared in terms of computational overhead and compression of the computational grid, using in addition either local or global time stepping strategies. We preliminarily conclude that the multiresolution techniques yield improved memory compression and gain in CPU time with respect to the adaptive mesh renement method.

Parallelisation of Multiscale-Based Grid Adaptation using Space-Filling Curves

a l'aide des courbes remplissantes. Abstract. The concept of fully adaptive multiscale nite volume schemes has been developed and investigated during the past decade. By now it has been successfully employed in numerous applications arising in engineering. In order to perform 3D computations for complex geometries in reasonable CPU time, the underlying multiscale-based grid adaptation strategy has to be parallelised via MPI for distributed memory architectures. In view of a proper scaling of the computational performance with respect to CPU time and memory, the load of data has to be well-balanced and communication between processors has to be minimised. This has been realised using space-lling curves.

Unsteady Overset Simulation of Rotor-Airframe Interaction

Inviscid, three-dimensional, unsteady e ow computational results for the aerodynamic interaction between a rotor and a hemispherical-cylinder airframe are presented. A multiblock strategy is used, with separate grids for the fuselage and the rotor blades. An overset grid methodology is used to communicate the e ow information between the grids. In each grid a second-order temporally accurate, e fth-order spatially accurate, implicit e nite volume scheme is employed to solve the Euler equations. The vortical wake generated by the blade is captured from e rst principles. The vorticity patterns that impinge on the airframe are analyzed, and the surface-pressure distributions on the airframe are compared with experimental values.

Lessons from LESFOIL Project on Large-Eddy Simulation of Flow Around an Airfoil

A synopsis of important results obtained by seven partners in the Brite ‐Euram project LESFOIL sponsored by the European Union is presented. This project was devoted to assessing the feasibility of large-eddy simulations (LES)forthecomputationofthee owaroundanairfoil.Asatestcase,theAerospatialeA-airfoilat Re=2£10 6 and 13.3-deg angle of attack was chosen. All partners performed simulations of this e ow with various methods, most of which employed e nite volume schemes and various subgrid-scale models of eddy-viscosity type. The key results of theindividual partners’ publicationsarepresented ina comparativeway.Itis demonstratedthatthiscone guration with realistic conditions for aeronautical e ows is extremely demanding for LES. One reason is the need to achieve a threshold resolution for the adequate discretization of the attached and mildly separating boundary layer. The second and even more demanding requirement is the need for an adequate treatment of transition, which in the present computations was achieved by increased resolution.

A fully coupled numerical technique for 2D bed morphology calculations

AbstractA fully coupled two‐dimensional subcritical and/or supercritical, viscous, free‐surface flow numerical model is developed to calculate bed variations in alluvial channels. Vertically averaged free‐surface flow equations in conjunction with sediment transport equation are numerically solved using an explicit finite‐volume scheme using transformed grid in order to handle complex geometry fluvial problems. Convergence is accelerated with use of a multi‐grid technique. Firstly the capabilities of the proposed method are demonstrated by analyzing subcritical and supercritical hydrodynamic flows. Thereafter, an analysis of one‐ and two‐dimensional flows is performed referring to aggradation and scouring. For all reported test cases the computed results compare reasonably well with measurements as well as with other numerical solutions. The method is stable, reliable and accurate handling a variety of sediment transport equations with rapid changes of sediment transport at the boundaries. Copyright © 2002 John Wiley & Sons, Ltd.

Nonequilibrium Planar Interface Model for Solidification of Semitransparent Radiating Materials

A nonequilibrium solidie cation model for semitransparent materials is presented. Consideration is given to a planar layer of emitting, absorbing, and scattering medium subject to radiative and convective cooling. The enthalpy method is used to formulate the phase-change problem together with radiative transfer equation taking into account internal emitting, absorbing, and scattering. A planar interface nonequilibrium solidie cation is assumed with crystalline phase nucleated on the surface at a given nucleation temperature, which may be signie cantly lower than the equilibrium melting temperature of the material. A linear kinetics relationship is introduced to correlate the unknown solidie cation temperature to the interface velocity. A fully implicit e nite volume scheme is used to solve the problem with the solidie cation interface tracked by a modie ed interface tracking method. Theradiative transfer equation issolved using the discreteordinates method. Internal radiation enhances the latent heat removal and thus leads to a higher interface velocity and a larger melt undercooling. Optical thickness and the conduction-radiation parameter are two important parameters that affect the solidie cation process. In the presence of external convective cooling, effect of internal radiation is small in the early stage of solidie cation.

Modeling of Three-Dimensional Viscous Compressible Turbomachinery Flows Using Unstructured Hybrid Grids

An advanced numerical model for the simulation of steady and unsteady viscous compressible e ows for turbomachinery applications is described. The compressible Favre-averaged Navier ‐Stokes equations are used together with a one-equation turbulence model. The e ow domain is discretized using unstructured hybrid grids that can contain a mixture of hexahedral, pentahedral, tetrahedral, and triangular prismatic cells. The e ow equations are discretized using a node-centered e nite volume scheme that relies on representing the mesh using an edge-based data structure. A dual time stepping technique is applied to a point implicit formulation so that time accuracy can be maintained with large Courant ‐Friedrichs‐Lewy numbers. Nonree ecting boundary conditions are applied at the ine ow and oute ow boundaries to prevent any spurious ree ections of the outgoing waves. The model was validated against measured data for two cases. Radial proe les of pressure and temperature rise were determined from the steady e ow analysis of a rig fan blade, and these were found to be in very good agreement with the measured quantities. A rotor/stator interaction was studied next. Detailed comparisons were carried out against measured steady and unsteady e ow data and good agreement was obtained in all cases.

Numerical Simulation of Transonic Flow over Wing-Mounted Twin-Engine Transport Aircraft

A numerical method has been developed for computing the e owe eld around advanced transport aircraft with wing-mounted nacelles. The method is based on a multiblock point-matched grid-generation approach combined with zonal solving strategy for complex e owe eld. In this study the e owe eld is divided into a number of nonoverlapped blocks by a cutout technique. H-type grids are generated independently in each block using an elliptic grid-generation method, in which the control of the grid quality is accomplished by the forcing-function technique of Hilgenstock. The e owe eld is simulated by solving the Euler equations. An explicit three-stage Runge ‐Kutta algorithm based on the Jameson’ s e nite volume scheme for the Euler equations has been developed that is applied to the multiregion H-type grids. The present method has been applied to isolated powered engine nacelles and complex transport aircrafts consisting of low-wing/fuselage with wing-mounted pylon/nacelles. On the wing surfaces the viscous effects are simulated by the employment of the viscous/inviscid interaction (VII) technique of two-dimensional strip boundary layer. In this study the boundary-layer program uses an integral method to calculate turbulent boundary layers. With the concept of an equivalent inviscid e ow, the model of blowing velocity is employed in the VII technique. The effect of the boundary layeron the outer inviscid e ow is represented through a transpiration boundaryconditionderived from theboundary-layerparameters.Themain benee tofthistreatment isthatthegridisgeneratedonlyonceinoverallcomputingprocedure.Computationalresultsandcomparisonswith experimental data are presented. The good agreement indicates that the present method is effective in predicting the e ows about powered engine nacelles and/or complex transport aircrafts.

Flow Simulation Using Generalized Static and Dynamic Grids

A new approach is presented for a e ow simulation system using generalized grids. In a generalized grid, the physical domain of interest is decomposed into cells with an arbitrary number of edges or faces. The grid can be of structured, unstructured, or hanging node type or an arbitrary combination of the types. A cell-face-based data structure is used to storethegrid information. A e ow simulation system is developed forgeneralized gridsthat can handle static and dynamic grids. The full Navier ‐Stokes equations, in the integral form, are taken as the relations that govern the e uid e ow. A cell-centered e nite volume scheme is developed for solving these governing equations. The numerical e ux across the cell faces is evaluated by an upwind scheme based on Roe’ s approximate Riemann solver. A higher-order scheme is formulated by utilizing Taylor’ s series expansion and Green’ s theorem. Limiter functions are used to preserve monotonocity. The skin-friction coefe cient is used to evaluate the accuracy of the limiterfunctions. Thegeneralized minimalresidualmethod isutilized tosolvethesparselinearsystem ofequations resulting from thelinearization of the e ux vectors. TheSpalart ‐Allmaras one-equation turbulence model has been implemented for the generalized grids and is used to evaluate the turbulent viscosity. For dynamically moving bodies, the equations of classical mechanics are used to predict the trajectory based on the external aerodynamic and body forces. A variety of computational examples are presented to demonstratethe wide rangeof applications, and the results are compared with experimental data whenever available.

Convergence rate of a finite volume scheme for a two dimensional convection-diffusion problem

In this paper, a class of cell centered nite volume schemes, on general unstructured meshes, for a linear convection-diusion problem, is studied. The convection and the diusion are respectively approximated by means of an upwind scheme and the so called diamond cell method (4). Our main result is an error estimate of order h, assuming only the W 2;p (for p> 2) regularity of the continuous solution, on a mesh of quadrangles. The proof is based on an extension of the ideas developed in (12). Some new diculties arise here, due to the weak regularity of the solution, and the necessity to approximate the entire gradient, and not only its normal component, as in (12).

Finite volume schemes for a nonlinear hyperbolic equation. Convergence towards the entropy solution and error estimate

In this paper, we study some nite volume schemes for the nonlinear hyperbolic equation ut(x;t )+d ivF (x;t;u(x;t)) = 0 with the initial condition u02 L 1 (R N ). Passing to the limit in these schemes, we prove the existence of an entropy solution u2 L 1 (R N R+). Proving also uniqueness, we obtain the convergence of the nite volume approximation to the entropy solution in L p (R N R+), 1 p +1. Furthermore, if u02 L 1 BVloc(R N ), we show that u2 BVloc(R N R+), which leads to an \h 1 4 error estimate between the approximate and the entropy solutions (where h denes the size

Shape Optimizing Nacelle near Flat-Plate Wing Using Multiblock Sensitivity Analysis

A major design task in reducing the overall aircraft drag is the integration of the engine nacelles and the airframe. With this impetus, a methodology was demonstrated to optimize nacelle shapes with and without the presence of a e at-plate nearby to account for the wing interference. Overly simplie ed shapes notwithstanding, this process requires multiblock grids not only for its aerodynamic analysis, but also for its optimization. Although the former is a standard practice, the latter has only recently been possible for the gradient-based optimizations with the development of the sensitivity analysis with domain decomposition scheme. The analyses were obtained by solving the three-dimensional, compressible, thin-layer Navier ‐ Stokes equations using an implicit, upwind-biased, e nite volume scheme. In addition to demonstrating the present method’ s suitability for automated shape optimization of interfering aircraft components, such as a nacelle and a wing, the results verie ed two important issues. First, accounting for the aerodynamic mutual interference between components in close proximity manifested itself in a shape different than that obtained when a component was assumed to be isolated. Secondly, even for isolatedcomponent designs, neglecting the viscous effects compromised not only the e ow physics but also the optimized shapes.