Order Accurate Finite Volume Method Research Articles

Active Flux Methods for Hyperbolic Systems Using the Method of Bicharacteristics

The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method is fully discrete and has a compact stencil in space and time. An important component of Active Flux methods is the evolution formula for the update of the point values. A previously proposed exact evolution formula for acoustics is reviewed and used to construct an Active Flux method for the two-dimensional Maxwell’s equation. Furthermore, the method of bicharacteristics is discussed as a methodology for the derivation of truly multidimensional approximative evolution operators that can be used for the evolution of point values in Active Flux methods. We study accuracy and stability of the resulting methods for acoustics and compare with the Active Flux method that uses the exact evolution operator. Finally, we used the method of bicharacteristics to derive Cartesian grid Active Flux methods for the linearised and nonlinear Euler equations. Numerous test computations illustrate the performance of these new Active Flux methods.

The Cartesian Grid Active Flux Method with Adaptive Mesh Refinement

We present the first implementation of the Active Flux method on adaptively refined Cartesian grids. The Active Flux method is a third order accurate finite volume method for hyperbolic conservation laws, which is based on the use of point values as well as cell average values of the conserved quantities. The resulting method has a compact stencil in space and time and good stability properties. The method is implemented as a new solver in ForestClaw, a software for parallel adaptive mesh refinement of patch-based solvers. On each Cartesian grid patch the single grid Active Flux method can be applied. The exchange of data between grid patches is organised via ghost cells. The local stencil in space and time and the availability of the point values that are used for the reconstruction, leads to an efficient implementation. The resulting method is third order accurate, conservative and allows the use of subcycling in time.

A quantification method for numerical dissipation in quasi-DNS and under-resolved DNS, and effects of numerical dissipation in quasi-DNS and under-resolved DNS of turbulent channel flows

LES for industrial applications with complex geometries is mostly characterised by: a) a finite volume CFD method using a non-staggered arrangement of the flow variables and second order accurate spatial and temporal discretisation schemes, b) an implicit top-hat filter, where the filter length is equal to the local computational cell size, and c) eddy-viscosity type LES models. LES based on these three main characteristics is indicated as industrial LES in this paper.It becomes increasingly clear that the numerical dissipation in CFD codes typically used in industrial applications with complex geometries may inhibit the predictive capabilities of explicit LES. Therefore, there is a need to quantify the numerical dissipation rate in such CFD codes. In this paper, we quantify the numerical dissipation rate in physical space based on an analysis of the transport equation for the mean turbulent kinetic energy. Using this method, we quantify the numerical dissipation rate in a quasi-Direct Numerical Simulation (DNS) and in under-resolved DNS of, as a basic demonstration case, fully-developed turbulent channel flow. With quasi-DNS, we indicate a DNS performed using a second order accurate finite volume method typically used in industrial applications. Furthermore, we determine and explain the trends in the performance of industrial LES for fully-developed turbulent channel flow for four different Reynolds numbers for three different LES mesh resolutions. The presented explanation of the mechanisms behind the observed trends is based on an analysis of the turbulent kinetic energy budgets.The presented quantitative analyses demonstrate that the numerical errors in the industrial LES computations of the considered turbulent channel flows result in a net numerical dissipation rate which is larger than the subgrid-scale dissipation rate.No new computational methods are presented in this paper. Instead, the main new elements in this paper are our detailed quantification method for the numerical dissipation rate, the application of this method to a quasi-DNS and under-resolved DNS of fully-developed turbulent channel flow, and the explanation of the effects of the numerical dissipation on the observed trends in the performance of industrial LES for fully-developed turbulent channel flows.

Turbulent convection and thermal radiation in a cuboidal Rayleigh–Bénard cell with conductive plates

The interactions between turbulent convection and surface-to-surface radiation are investigated by means of direct numerical simulations (DNS) performed for a cuboidal Rayleigh–Bénard cell bounded by horizontal plates of finite thickness and with high and low conductivities. The simulations are conducted for the Prandtl number Pr=0.7, Rayleigh numbers ranging from Ra=3.5×105 to Ra=6.3×107, radiation numbers between Nr=0.0008 and Nr=0.0026, and temperature ratios between Θ=29 and Θ=45. Additionally, blackbody radiation is considered, and the side walls are assumed to be transparent to radiation but adiabatic for conduction. The DNS are performed with a well validated fourth order accurate finite volume method. Analysing the DNS data it is shown that radiation renders the bulk more uniform by enlarging the large scale circulations and damps turbulence intensities by decreasing the buoyancy force. How these effects depend on Ra and the plates’ diffusivity is also discussed. The temperatures of the cold plumes are affected equally by radiation, while those of the hot plumes are affected differently, depending on their location. Radiation affects the temperature of the heat sources by cooling the heating plates and heating the cooling plates. The spatial fluctuations at the interfaces are decreased or increased by radiation, depending on the plates’ diffusivity. The temporal fluctuations are not affected by radiation when high conductivity plates are employed. The radiative heat flux contributes considerably to the total one, particularly for low conductivity plates and low Ra. Heat radiation also initiates a so-called convective drop. It is found that in regions where the plumes emerge, only the radiative heat flux and its intensity are responsible for this drop, while in regions where the plumes arrive, the convective heat transported by the arriving plumes contributes as well. Small variations in the temperature distribution at the interfaces do not affect irradiation, which implies the possibility of further simplifications and a decrease in the computational cost. The convective and radiative heat fluxes are anticorrelated for low conductivity plates and uncorrelated in space for high conductivity plates. Furthermore, it is shown that the radiative heat transfer increases with the control parameters for radiation, yet the increase is more sensitive to variations of the radiation number than the temperature ratio. On the other hand, the convective heat transfer and the effective temperature difference decrease whenever any of the control parameters for radiation increase. The meaning of the control parameters for radiation is discussed and the dependencies are expressed with scaling functions.

Free Surface Flow in a Trench Channel Using 3-D Finite Volume Method

In order to simulate a free surface flow in a trench channel, a three-dimensional incompressible unsteady Reynolds-averaged Navier-Stokes (RANS) equations are closed with the <TEX>${\kappa}-{\epsilon}$</TEX> model. The artificial compressibility (AC) method is used. Because the pressure fields can be coupled directly with the velocity fields, the incompressible Navier-Stokes (INS) equations can be solved for the unknown variables such as velocity components and pressure. The governing equations are discretized in a conservation form using a second order accurate finite volume method on non-staggered grids. In order to prevent the oscillatory behavior of computed solutions known as odd-even decoupling, an artificial dissipation using the flux-difference splitting upwind scheme is applied. To enhance the efficiency and robustness of the numerical algorithm, the implicit method of the Beam and Warming method is employed. The treatment of the free surface, so-called interface-tracking method, is proposed using the free surface evolution equation and the kinematic free surface boundary conditions at the free surface instead of the dynamic free surface boundary condition. AC method in this paper can be applied only to the hydrodynamic pressure using the decomposition into hydrostatic pressure and hydrodynamic pressure components. In this study, the boundary-fitted grids are used and advanced each time the free surface moved. The accuracy of our RANS solver is compared with the laboratory experimental and numerical data for a fully turbulent shallow-water trench flow. The algorithm yields practically identical velocity profiles that are in good overall agreement with the laboratory experimental measurement for the turbulent flow.

Numerical determination of permeability of Al–Cu alloys using 3D geometry from X-ray microtomography

In the present study, the permeabilities of Al–15·5 wt-%Cu and Al–19·5 wt-%Cu have been determined numerically by solving the full Navier–Stokes equations on 3D geometries of interdendritic channels obtained by X-ray microtomography. The samples for X-ray microtomography imaging were sectioned from different heights in directionally solidified cylinders in order to obtain a variety of microstructures for each composition. The flow has been solved using a second order accurate finite volume method approach. The effect of employing uniform or unstructured mesh on the calculated permeability has been studied. The Marching–Cube triangulation method was used to extract accurate surfaces for unstructured volume meshing. Calculated values of permeability for this range of solid fractions show partial agreement with previous experiments. Observed deviations are analysed and ultimately attributed to experimental error associated with the difficulties of measuring permeability.

The enhanced cooling of heated blocks mounted on the wall of a plane channel filled with a porous medium

This is a two dimensional numerical simulation of the laminar air forced convection cooling of six blocks mounted on the lower wall of a plane horizontal channel filled (or not filled) with a porous medium. The blocks are equally spaced and heated by a uniform volumetric heat generation. The objective of the study is the determination of the enhanced cooling of the blocks when the channel is filled with a porous medium. The problem is modelled by the continuity, momenta and energy equations with their appropriate initial and boundary conditions. When the channel is filled with a porous medium, a Darcy-Forchheimer-Brinkman flow model is used to describe the flow field. The model equations are numerically solved by a second order accurate finite volume method. The results show that the flow field of the channel filled with a porous medium is very different from that of the channel without the porous matter. A major difference is the appearance of circulating vortices between the blocks when the porous matter is inexistent. This flow pattern difference favours a convective heat transfer enhancement of the flow in the channel filled with the porous matter. Moreover, the effective thermal conductivity of the considered porous medium is higher than that of the fluid. These two factors lead to a considerably better cooling of the blocks mounted in the channel filled with the porous matter. Thus, the use of porous media when possible is recommended because it enhances the cooling of heated blocks mounted in channels.

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.

Influence of curvature and torsion on turbulent flow in helically coiled pipes

Fully-developed, statistically steady turbulent flow in straight, curved and helically coiled pipes is studied by means of direct numerical simulation for a Reynolds number of Reτ=230. The incompressible Navier–Stokes equations, written in orthogonal helical coordinates, are integrated numerically by means of a second order accurate finite volume method. It is shown that pipe curvature which induces a secondary flow has a strong effect on the flow quantities. Turbulence is significantly inhibited by streamline curvature and the flow almost relaminarizes for high values of the curvature parameter (κ=0.1). The torsion effect is weaker than the curvature effect. Nevertheless, it cannot be neglected. It influences the secondary flow induced by pure curvature and leads to an increase in fluctuating kinetic energy and dissipation rate.

DNS OF 2D TURBULENT FLOW AROUND A SQUARE CYLINDER

Two-dimensional ‘turbulent’ flow around a rectangular cylinder has been simulated at Re=10,000 using a sixth- order-accurate finite volume method for the discretization of convection and diffusion. The spatial discretization consists of a combination of a seventh- order upwind-biased method for the convective terms and an eighth-order central method for the diffusive terms, discretized on a stretched and staggered grid. To cope with the stretching of the grid, Lagrange interpolations are used. The method applied to obtain a boundary condition for the velocity in the x-direction at the outflow boundary is shown not to affect the flow in the interior of the computational domain in a way that is visible in various snapshots of the vorticity field. The variation in the velocity in the x-direction with time is itself found to be relatively small near the outflow boundary. Several turbulence statistics have been gathered from a simulation of the flow developed during 77 dimensionless time units. Snapshots of the vorticity field of the developed flow show the presence of a vortex-street- like structure. Typical 2D turbulent behaviour, such as the appearance of monopolar, dipolar and tripolar vortices due to the amalgamation of vorticity in the wake and the x−1/2 scaling of the velocity defect in the wake, has been obtained. © 1997 John Wiley & Sons, Ltd.

DNS OF 2D TURBULENT FLOW AROUND A SQUARE CYLINDER

Two-dimensional ‘turbulent’ flow around a rectangular cylinder has been simulated at Re=10,000 using a sixth- order-accurate finite volume method for the discretization of convection and diffusion. The spatial discretization consists of a combination of a seventh- order upwind-biased method for the convective terms and an eighth-order central method for the diffusive terms, discretized on a stretched and staggered grid. To cope with the stretching of the grid, Lagrange interpolations are used. The method applied to obtain a boundary condition for the velocity in the x-direction at the outflow boundary is shown not to affect the flow in the interior of the computational domain in a way that is visible in various snapshots of the vorticity field. The variation in the velocity in the x-direction with time is itself found to be relatively small near the outflow boundary. Several turbulence statistics have been gathered from a simulation of the flow developed during 77 dimensionless time units. Snapshots of the vorticity field of the developed flow show the presence of a vortex-street- like structure. Typical 2D turbulent behaviour, such as the appearance of monopolar, dipolar and tripolar vortices due to the amalgamation of vorticity in the wake and the x−1/2 scaling of the velocity defect in the wake, has been obtained. © 1997 John Wiley & Sons, Ltd.