Accurate Finite Volume Research Articles

A fourth order accurate finite volume scheme for numerical simulations of turbulent Rayleigh–Bénard convection in cylindrical containers

A finite volume scheme, which is based on fourth order accurate central differences in spatial directions and on a hybrid explicit/semi-implicit time stepping scheme, was developed to solve the incompressible Navier–Stokes and energy equations on cylindrical staggered grids. This includes a new fourth order accurate discretization of the velocity and temperature fields at the singularity of the cylindrical coordinate system and a new stability condition [J. Appl. Numer. Anal. Comput. Math. 1 (2004) 315–326]. The method was applied in direct numerical simulations of turbulent Rayleigh–Bénard convection for different Rayleigh numbers Ra = 10 γ , γ = 5 , … , 8 , in wide cylinders with the aspect ratios a ≡ H / R = 0.2 and a = 0.4 (where R denotes the radius and H – the height of the cylinder). To cite this article: O. Shishkina, C. Wagner, C. R. Mecanique 333 (2005).

UNSTEADY FLOW COMPUTATIONS OVER MOVING BODY USING DYNAMIC MESHES

An efficient implicit unstructured grid algorithm for solving unsteady inviscid compressible flows over moving body employing an Arbitrary Lagrangian Eulerian formulation is presented. In the present formulation, the time discretization is performed using a second-order accurate 3-point time integration scheme and the upwind-biased space discretization using second-order accurate finite volume formulation with Venkatakrishnan limiter. The face-velocities of the control volumes are computed using Geometric Conservation Laws. The nonlinear system arising from the implicit formulation is solved using an ILU preconditioned Newton–Krylov iteration at every time step. The computed results for two test cases involving harmonically oscillating NACA0012 airfoil are presented in order to demonstrate the efficacy of the present solver.

Effects of Wall Rotation on Heat Transfer to Annular Turbulent Flow: Outer Wall Rotating

AbstractSimulations were conducted for air flowing upward in a vertical annular pipe with a rotating outer wall. Simulations concentrated on the occurrence of laminarization and property variations for high heat flux heat transfer. The compressible filtered Navier-Stokes equations were solved using a second-order accurate finite volume method. Low Mach number preconditioning was used to enable the compressible code to work efficiently at low Mach numbers. A dynamic subgrid-scale stress model accounted for the subgrid-scale turbulence. When the outer wall rotated, a significant reduction of turbulent kinetic energy was realized near the rotating wall and the intensity of bursting appeared to decrease. This modification of the turbulent structures was related to the vortical structure changes near the rotating wall. It has been observed that the wall vortices were pushed in the direction of rotation and their intensity increased near the nonrotating wall. The consequent effect was to enhance the turbulent kinetic energy and increase the Nusselt number there.

Large eddy simulation of turbulent forced gas flows in vertical pipes with high heat transfer rates

Large eddy simulation (LES) of vertical turbulent pipe flows with significant property variations has been performed to investigate the effects of high heat fluxes on the turbulent structures and transport. The Cartesian-based, compressible filtered Navier–Stokes equations were solved using a second-order accurate finite volume method. Low Mach number preconditioning was used to enable the compressible code to perform efficiently at low Mach numbers. A dynamic subgrid-scale stress model accounted for the subgrid-scale turbulence. In this study, the simulations were designed to simulate the experiments of Shehata and McEligot with three different near-constant heat fluxes. Step-periodic boundary conditions based on a quasi-developed assumption were used. The predicted integral parameters and mean velocity and temperature profiles agreed well with the experimental data. The fluid structures have been distorted due to high heat fluxes leading to significant property variations in the near wall region. The results showed that strong heating resulted in remarkable reductions of turbulent intensities, shear stresses, and turbulent heat flux. Apparent “laminarization” of the flow has been observed.

An accurate moving grid Eulerian Lagrangian localized adjoint method for solving the one‐dimensional variable‐coefficient ADE

AbstractAn accurate finite‐volume Eulerian Lagrangian localized adjoint method (ELLAM) is presented for solving the one‐dimensional variable coefficients advection dispersion equation that governs transport of solute in porous medium. The method uses a moving grid to define the solution and test functions. Consequently, the need for spatial interpolation, or equivalently numerical integration, which is a major issue in conventional ELLAM formulations, is avoided.After reviewing the one‐dimensional method of ELLAM, we present our strategy and detailed calculations for both saturated and unsaturated porous medium. Numerical results for a constant‐coefficient problem and a variable‐coefficient problem are very close to analytical and fine‐grid solutions, respectively. The strength of the developed method is shown for a large range of CFL and grid Peclet numbers. Copyright 2004 John Wiley & Sons, Ltd.

High-order finite volume methods for viscoelastic flow problems

In this paper accurate and stable finite volume schemes for solving viscoelastic flow problems are presented. Two contrasting finite volume schemes are described: a hybrid cell-vertex scheme and a pure cell-centred counterpart. Both schemes employ a time-splitting algorithm to evolve the solution through time towards steady state. In the case of the hybrid scheme, a semi-implicit formulation is employed in the momentum equation, based on the Taylor–Galerkin approach with a pressure-correction step to enforce incompressibility. The basis of the pure finite volume approach is a backward Euler scheme with a semi-Lagrangian step to treat the convection terms in the momentum and constitutive equations. Two distinct finite volume schemes are presented for solving the systems of partial differential equations describing the flow of viscoelastic fluids. The schemes are constructed to be second-order accurate in space. The issue of stability is also addressed with respect to the treatment of convection. Numerical examples are presented illustrating the performance of these schemes on some steady and transient problems that possess analytical solutions.

Stable and accurate hybrid finite volume methods based on pure convexity arguments for hyperbolic systems of conservation law

This exploratory work tries to present first results of a novel approach for the numerical approximation of solutions of hyperbolic systems of conservation laws. The objective is to define stable and “reasonably” accurate numerical schemes while being free from any upwind process and from any computation of derivatives or mean Jacobian matrices. That means that we only want to perform flux evaluations. This would be useful for “complicated” systems like those of two-phase models where solutions of Riemann problems are hard, see impossible to compute. For Riemann or Roe-like solvers, each fluid model needs the particular computation of the Jacobian matrix of the flux and the hyperbolicity property which can be conditional for some of these models makes the matrices be not R -diagonalizable everywhere in the admissible state space. In this paper, we rather propose some numerical schemes where the stability is obtained using convexity considerations. A certain rate of accuracy is also expected. For that, we propose to build numerical hybrid fluxes that are convex combinations of the second-order Lax–Wendroff scheme flux and the first-order modified Lax–Friedrichs scheme flux with an “optimal” combination rate that ensures both minimal numerical dissipation and good accuracy. The resulting scheme is a central scheme-like method. We will also need and propose a definition of local dissipation by convexity for hyperbolic or elliptic-hyperbolic systems. This convexity argument allows us to overcome the difficulty of nonexistence of classical entropy-flux pairs for certain systems. We emphasize the systematic feature of the method which can be fastly implemented or adapted to any kind of systems, with general analytical or data-tabulated equations of state. The numerical results presented in the paper are not superior to many existing state-of-the-art numerical methods for conservation laws such as ENO, MUSCL or central scheme of Tadmor and coworkers. The interest is rather the systematic feature of the method and its very fast implementation for prototypying and fluid model validation. In this context, the Rusanov scheme is often used; the present approach here gives far better results.

On the application of congruent upwind discretizations for large eddy simulations

Upwind schemes were judged inappropriate for performing accurate large eddy simulations of turbulent flow owing to the artificial dissipation that is present at high wavenumbers of the energy content. Such a conclusion has been drawn also from some results obtained by adopting Finite Difference schemes. The present paper illustrates the performances of some new Finite Volume upwind discretization of the convective terms in the case of the 1-D Burgers model equation while studying the effects of numerical discretization on several Sub-Grid Scales turbulence models, starting from the classical static and dynamic eddy viscosity models through the recent deconvolution-based ones. Basing on previously published papers, large eddy simulations along with a deconvolution-based procedure for de-filtering the evolving variable, have been originally developed and applied. It will be shown how the coherent application of the procedure allows us to develop high-order accurate Finite Volume upwind schemes, which maintain a good spectral resolution in the entire range of resolved scale. Such schemes can be candidate for performing accurate simulations of real turbulence.

Large eddy simulation of heated vertical annular pipe flow in fully developed turbulent mixed convection

The goal of the present study was to perform a large eddy simulation of vertical turbulent annular pipe flow under conditions in which fluid properties vary significantly, and to investigate the effects of buoyancy on the turbulent structures and transport. Isoflux wall boundary conditions with low and high heating are imposed. The compressible filtered Navier–Stokes equations are solved using a second order accurate finite volume method. Low Mach number preconditioning is used to enable the compressible code to work efficiently at low Mach numbers. A dynamic subgrid-scale stress model accounts for the subgrid-scale turbulence. Comparisons were made with available experimental data. The results showed that the strong heating and buoyant force caused distortions of the flow structure resulting in reduction of turbulent intensities, shear stress, and turbulent heat flux, particularly near the wall.

A space?time analysis of blood flow in a 3D vessel with multiple aneurysms

In this study, 3D unsteady flow dynamics in an arterial vessel with two asymmetric aneurysms in series have been numerically investigated under pulsatile flow conditions for a full cycle of period T. The non-linear partial differential equations governing the conservation of mass and momentum of viscous incompressible fluid have been numerically analysed by a time accurate cell centered finite volume method in implicit Euler time marching setting. The influence of Reynolds' number (Re), Strouhal's number (St), and degree of dilations (δ1 1, δ2 1, δ1 2, δ2 2) on wall shear stress (WSS), wall pressure (WP) and velocities (u, v, w) have been investigated. While the WSSs increase with either increasing St or increasing Re, WPs are seen to get lowered. During the systolic phase high WPs are seen at the distal end of aneurysms, especially at the distal tip of the larger aneurysm. Decreasing St is seen to delay the flow separation process but the vortex shedding with 3D features is always noticed towards the end of the diastolic phase of the flow cycle.

Large Eddy Simulation of Constant Heat Flux Turbulent Channel Flow With Property Variations: Quasi-Developed Model and Mean Flow Results

Turbulent planar channel flow has been computed for uniform wall heating and cooling fluxes strong enough to cause significant property variations using large eddy simulation. Channels with both walls either heated or cooled were considered, with wall-to-bulk temperature ratios as high as 1.5 for the heated case, and as low as 0.56 for the cooled case. An implicit, second order accurate finite volume scheme was used to solve the time dependent filtered set of equations to determine the large eddy motion, while a dynamic subgrid-scale model was used to account for the subgrid scale effects. Step-periodicity was used based on a quasi-developed assumption. The effects of strong heating and cooling on the flow were investigated and compared with the results obtained under low heating conditions.

Numerical Simulation of Unsteady Turbulent Flow Around Maneuvering Prolate Spheroid

A three-dimensional Reynolds averaged Navier-Stokes method for unsteady turbulent flow around a maneuvering vehicle was developed and applied to a model problem concerning an extreme case of submarine maneuvers. A body force term is added in the momentum equations to take into account the inertial motion in the body-fixed coordinate system. The Spalart and Allmaras turbulence model is employed for turbulence closure. An artificial compressibility is introduced into the continuity equation for velocity-pressure coupling. The governing equations are discretized by second-order accurate finite volume method in space and second-order accurate backward scheme in time. The computational results are analyzed with global and local quantities and validated by comparison with experimental data. Overall, the present method performs quite well in predicting the unsteady flow phenomena associated with a maneuvering prolate spheroid, and the results compare well with available experimental data

Unsteady aerodynamic calculations using three-dimensional Euler equations on unstructured dynamic grids

AbstractThis paper presents an algorithm to solve the three-dimensional unsteady Euler equations on unstructured tetrahedral meshes using a dynamic mesh algorithm. The driving algorithm is an upwind biased implicit second order accurate cell-centered finite volume scheme. The spatial discretisation technique involves a naturally dissipative flux-split approach that accounts for the local wave propagation characteristics of the flow and captures shock waves sharply. A continuously differentiable flux limiter has been employed to eliminate the spurious oscillations near shock waves, generally arising in calculations involving upwind biased schemes. The temporal discretisation is also second order accurate and uses a Newton linearisation for unsteady calculations. To calculate time dependent flows a dynamic mesh algorithm has been implemented in which the mesh is moved to conform to the instantaneous position of the body by modelling each edge of each cell by a spring. The paper presents a description of the solver and the grid movement algorithm along with results and comparison that assess their capabilities.

A numerical study of an unsteady laminar flow in a doubly constricted 3D vessel

AbstractUnsteady flow dynamics in doubly constricted 3D vessels have been investigated under pulsatile flow conditions for a full cycle of period T. The coupled non‐linear partial differential equations governing the mass and momentum of a viscous incompressible fluid has been numerically analyzed by a time accurate Finite Volume Scheme in an implicit Euler time marching setting. Roe's flux difference splitting of non‐linear terms and the pseudo‐compressibility technique employed in the current numerical scheme makes it robust both in space and time. Computational experiments are carried out to assess the influence of Reynolds' number and the spacing between two mild constrictions on the pressure drop across the constrictions. The study reveals that the pressure drop across a series of mild constrictions can get physiologically critical and is also found to be sensitive both to the spacing between the constrictions and the oscillatory nature of the inflow profile. The flow separation zone on the downstream constriction is seen to detach from the diverging wall of the constriction leading to vortex shedding with 3D features earlier than that on the wall in the spacing between the two constrictions. Copyright © 2002 John Wiley & Sons, Ltd.

Laminar and turbulent Rayleigh–Bénard convection in a perfectly conducting cubical cavity

This paper discusses flow structures and heat transfer rates generated by Rayleigh–Bénard convective motions of a Boussinesq fluid with a Prandtl number of 0.7 in a perfectly conducting cubical cavity. Complete numerical simulations of laminar flows were conducted in the range of Rayleigh numbers 7×10 3⩽ Ra⩽10 5. The large-eddy simulation (LES) technique was used for the simulations at two high Rayleigh numbers ( Ra=10 6 and 10 8). LES were carried out using a second-order accurate finite volume code with a dynamic localized one-equation subgrid-scale (SGS) model with constant SGS Prandtl number. In the laminar regime, two single roll structures and a four-roll structure in which the axis of each roll is perpendicular to one sidewall were found to be stable. LES of Rayleigh–Bénard convection in an infinite fluid layer were initially carried out and results were seen to be in agreement with direct numerical simulations (DNS) reported in the literature. At Ra=10 6 and 10 8, the instantaneous velocity and temperature fields present strong fluctuations with respect to the time-averaged flow field. The confining effect of the conductive lateral walls of the cavity generates, in the unsteady flows at Ra=10 6 and 10 8, persistent vertical currents near these walls. The recirculation of these ascending and descending flows towards the central region of the cavity produces large-scale organized rolling motions, which imprint the topology of the time-averaged flow field in form of two vortex ring structures located near the horizontal walls.

Third Order Accurate Large-Particle Finite Volume Method on Unstructured Triangular Meshes

The large-particle fluid in cell (FLIC) method, presented in 1960s, is a numerical method that can be applied to solve unsteady flow. The computational scheme consists of two steps for each time march step: First, intermediate values are calculated for the velocities and energy, taking into account only the effects of acceleration caused by pressure gradients; second, transport effects are calculated. In this paper, we present a third order large-particle finite volume method for unstructured triangular meshes. The key idea of this method is the reconstruction of weighted quadratic interpolation for flow variables, and the two-point Gauss quadrature formula is used on mesh edges. The computational results for two typical flows verify the accuracy of the method.

A Parallel 3D Unsteady Flow Analysis in an Asymmetrically Constricted Vessel

In this study, parallel computation of unsteady incompressible flow in an asymmetrically constricted 3D vessel has been presented. A time accurate cell centered finite volume method (FVM) in conjunction with pseudo-compressibility technique and Roe's flux difference splitting of nonlinear terms has been employed for solving the Navier-Stokes (NS) equations on the multiple instruction multiple data (MIMD) machine VPP700. The influence of Reynolds' number ( Re ) and the Strouhal number ( St ) on flow dynamic factors like wall pressure (WP), wall shear stress (WSS), central axis velocity (CAV), etc., have been analyzed. Three-dimensional (3D) features in the formation and detachment of separation zones, which are sensitive to both Re and St have been noticed on the diverging wall of the constriction.

A Nondiffusive Finite Volume Scheme for the Three-Dimensional Maxwell's Equations on Unstructured Meshes

We prove a sufficient CFL-like condition for the L2 -stability of the second-order accurate finite volume scheme proposed by Remaki for the time-domain solution of Maxwell's equations in heterogeneous media with metallic and absorbing boundary conditions. We yield a very general sufficient condition valid for any finite volume partition in two and three space dimensions. Numerical tests show the potential of this original finite volume scheme in one, two, and three space dimensions for the numerical solution of Maxwell's equations in the time-domain.

Coupled aeroelastic oscillations of a turbine blade row in 3D transonic flow

This paper presents the mutual time - marching method to predict the aeroelastic stability of an oscillating blade row in 3D transonic flow. The ideal gas flow through a blade row is governed by the time dependent Euler equations in conservative form which are integrated by using the explicit monotonous second order accurate Godunov-Kolgan finite volume scheme and moving hybrid H-O grid. The structure analysis uses the modal approach and 3D finite element dynamic model of blade. The blade movement is assumed as a linear combination of the first modes of blade natural oscillations with the modal coefficients depending on time. To demonstrate the capability and correctness of the method, two experimentally investigated test cases have been selected, in which the blades had performed tuned harmonic bending or torsional vibrations (The 1st and 4th standard configurations of the “Workshop on Aeroelasticity in Turbomachines” by Bolcs and Fransson, 1986). The calculated results of aeroelastic behaviour of the blade row (4th standard configuration), are presented over a wide frequency range under different start regimes of interblade phase angle.

Numerical Assessment of Error Estimators for Euler Equations

Grid convergence studies are conducted to assess four error estimators for their asymptotic behavior: explicit residual, solution reconstruction, Richardson extrapolation, and solution of the error equations. Their accuracy, reliability, and efficitivity to predict the true error are verified on the quasi-one-dimensional Euler equations solved by a second-order accurate finite volume method