Order Upwind Scheme Research Articles

An Improved Second-Order Finite-Volume Algorithm for Detached-Eddy Simulation Based on Hybrid Grids

AbstractA hybrid grid based second-order finite volume algorithm has been developed for Detached-Eddy Simulation (DES) of turbulent flows. To alleviate the effect caused by the numerical dissipation of the commonly used second order upwind schemes in implementing DES with unstructured computational fluid dynamics (CFD) algorithms, an improved second-order hybrid scheme is established through modifying the dissipation term of the standard Roe's flux-difference splitting scheme and the numerical dissipation of the scheme can be self-adapted according to the DES flow field information. By Fourier analysis, the dissipative and dispersive features of the new scheme are discussed. To validate the numerical method, DES formulations based on the two most popular background turbulence models, namely, the one equation Spalart-Allmaras (SA) turbulence model and the two equationk–ωShear Stress Transport model (SST), have been calibrated and tested with three typical numerical examples (decay of isotropic turbulence, NACA0021 airfoil at 60° incidence and 65° swept delta wing). Computational results indicate that the issue of numerical dissipation in implementing DES can be alleviated with the hybrid scheme, the resolution for turbulence structures is significantly improved and the corresponding solutions match the experimental data better. The results demonstrate the potentiality of the present DES solver for complex geometries.

Viriato: A Fourier–Hermite spectral code for strongly magnetized fluid–kinetic plasma dynamics

We report on the algorithms and numerical methods used in Viriato, a novel fluid–kinetic code that solves two distinct sets of equations: (i) the Kinetic Reduced Electron Heating Model (KREHM) equations (Zocco and Schekochihin, 2011) (which reduce to the standard Reduced-MHD equations in the appropriate limit) and (ii) the kinetic reduced MHD (KRMHD) equations (Schekochihin et al., 2009). Two main applications of these equations are magnetized (Alfvénic) plasma turbulence and magnetic reconnection. Viriato uses operator splitting (Strang or Godunov) to separate the dynamics parallel and perpendicular to the ambient magnetic field (assumed strong). Along the magnetic field, Viriato allows for either a second-order accurate MacCormack method or, for higher accuracy, a spectral-like scheme composed of the combination of a total variation diminishing (TVD) third order Runge–Kutta method for the time derivative with a 7th order upwind scheme for the fluxes. Perpendicular to the field Viriato is pseudo-spectral, and the time integration is performed by means of an iterative predictor–corrector scheme. In addition, a distinctive feature of Viriato is its spectral representation of the parallel velocity-space dependence, achieved by means of a Hermite representation of the perturbed distribution function. A series of linear and nonlinear benchmarks and tests are presented, including a detailed analysis of 2D and 3D Orszag–Tang-type decaying turbulence, both in fluid and kinetic regimes.

English

In the present work, the effect of nozzle angle (22.5º, 45º and 67.5º) on mixing time for jet mixing tanks with the various ratios of liquid height (H) to tank diameter (D), including 0.5, 1, and 1.5, are studied by using computational fluid dynamics (CFD). The results revealed that CFD model with standard k-epsilon is successfully employed to predict the concentration profiles and mixing time by using the fine mesh and second order upwind scheme. The simulated results showed that the different jet nozzle angles result in different flow patterns. The results also indicate that the mixing time is mainly a function of the jet potential core length. Moreover, the jet path length or jet centerline velocity (jet kinetic energy) is considered as the secondary effect on mixing time, which depends on the tank geometry. Key words: Computational fluid dynamics (CFD), jet, mixing, turbulence, k-epsilon model.

An upwind-based scheme for solving the oblique derivative boundary-value problem related to physical geodesy

Abstract The paper presents a novel original upwindbased approach for solving the oblique derivative boundary value problem by the finite volume method. In this approach, the oblique derivative boundary condition is interpreted as a stationary advection equation for the unknown disturbing potential. Its approximation is then performed by using the first order upwind scheme taking into account information from inflow parts of the finite volume boundary only. When the numerical scheme is derived, numerical simulations in 2D and 3D domains are performed and the experimental order of convergence of the proposed algorithm is studied. Moreover a comparison with a solution by the central scheme previously used for this kind of problem is performed. Finally we present numerical experiments dealing with the global and local gravity field modelling.

Stability, accuracy and numerical diffusion analysis of nodal expansion method for steady convection diffusion equation

The traditional finite difference method or finite volume method (FDM or FVM) is used for HTGR thermal-hydraulic calculation at present. However, both FDM and FVM require the fine mesh sizes to achieve the desired precision and thus result in a limited efficiency. Therefore, a more efficient and accurate numerical method needs to be developed. Nodal expansion method (NEM) can achieve high accuracy even on the coarse meshes in the reactor physics analysis so that the number of spatial meshes and computational cost can be largely decreased. Because of higher efficiency and accuracy, NEM can be innovatively applied to thermal-hydraulic calculation.In the paper, NEMs with different orders of basis functions are successfully developed and applied to multi-dimensional steady convection diffusion equation. Numerical results show that NEMs with three or higher order basis functions can track the reference solutions very well and are superior to second order upwind scheme and QUICK scheme. However, the false diffusion and unphysical oscillation behavior are discovered for NEMs. To explain the reasons for the above-mentioned behaviors, the stability, accuracy and numerical diffusion properties of NEM are analyzed by the Fourier analysis, and by comparing with exact solutions of difference and differential equation. The theoretical analysis results show that the accuracy of NEM increases with the expansion order. However, the stability and numerical diffusion properties depend not only on the order of basis functions but also on the parity of the order. The numerical experiments are carried out to validate the above conclusions, which provide some significant guides for the development of the new NEMs. It can be concluded that NEMs have great potential to solve thermal hydraulic problems effectively, and can be used in the engineering design code.

Upwind Based Parameter Uniform Convergence Analysis for Two Parametric Parabolic Convection Diffusion Problems by Moving Mesh Methods

AbstractA parabolic two parametric convection‐diffusion reaction problem is considered for the moving mesh error analysis. The continuous problem is discretized by the first order upwind scheme on a non uniform mesh. A curvature based error monitor function is proposed to generate the layer adapted mesh. It is proved that the numerical solution converges to the exact solution on the mesh obtained by the equidistribution of the proposed monitor function. The convergence is first order accurate. The present analysis generalizes the results obtained in earlier publications [8,9]. (© 2015 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Key factors in the use of DDES for the flow around a simplified car

The Ahmed car body represents a generic vehicle exhibiting key aspects of the 3D flow arising due to standard automobile designs. It is recognised to be a challenging test case for the turbulence modelling community; combining strong separation with a pair of counter-rotating vortices, which interact to produce a downstream recirculation region. In recent years this case has been extensively studied using a range of methods, with varying success. In general, conclusions have been made on the basis of the standard form of each model, while in the present work we focus on variants of the common Delayed Detached-Eddy Simulation (DDES) approach, in order to demonstrate its sensitivity to commonly varied aspects of its usage. We demonstrate that variations in the usage of a single approach can easily be of the order of those observed when using different approaches.Previous studies, reconfirmed here, indicate that the majority of standard single point closure turbulence models are unable to provide a satisfactory prediction of the recirculating flow region aft of the body. This holds regardless of mesh resolution, model selection or numerical scheme. These models under-predict levels of turbulence over the slanted back, leading to over-prediction of the size of the separation region. DDES can offer an improved prediction although, while better than URANS, the use of DDES in its standard form still falls short of equivalent results obtained from either wall modelled or wall resolved Large Eddy Simulation. In the present work we investigate four aspects of DDES in an attempt to identify mechanisms for improving DDES for this representative case: (1) the underlying RANS model, (2) mesh resolution, (3) numerical scheme and (4) the addition of turbulent fluctuations.We observe that with insufficient mesh resolution the DDES models produce worse results than the URANS models. While first order methods are inappropriate, the more commonly selected second order upwind scheme is also demonstrated to have substantial adverse impact. As mesh resolution is increased the influence of the underlying RANS model diminishes. In a zonal RANS–DDES approach, the domain was split in two just before the rear slant, and the upstream RANS solution is used to inform a DDES calculation via the superposition of synthetic fluctuations at the interface. This technique is demonstrated to substantially improve the prediction, whilst also reducing the overall simulation cost by virtue of a smaller domain size. The injection of synthetic fluctuations provides a more accurate level of turbulence at the onset of separation and thereby overcomes the lack of resolved turbulence in the initial separated shear layer.

A note on the stability of cut cells and cell merging

Embedded boundary meshes may have cut cells of arbitrarily small volume which can lead to stability problems in finite volume computations with explicit time stepping. We show that time step constraints are not as strict as often believed. We prove this in one dimension for linear advection and the first order upwind scheme. Numerical examples in two dimensions demonstrate that this carries over to more complicated situations. This analysis sheds light on the choice of time step when using cell merging to stabilize the arbitrarily small cells that arise in embedded boundary schemes.

A computational study of three numerical methods for some advection-diffusion problems

Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme [5], fourth order upwind scheme [5] and non-standard finite difference scheme (NSFD) [10]. We considered two test problems. The first test problem we considered has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries [16]. Many methods suffer from computational noise when modeling the second test problem. We compute some errors, namely L2 and L∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems. We then use an optimization technique [1] to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by numerical experiments.

Analysis of dust distribution in silo during axial filling using computational fluid dynamics: Assessment on dust explosion likelihood

In this study, the dust distribution in a silo during axial filling was modelled using a commercial computational fluid dynamics (CFD) code. The work focused on the dust concentration distribution in the silo, for evaluating the likelihood of a dust explosion in the silo. The simulation was conducted using a combination of renormalized (RNG) k-epsilon and discrete phase models, with standard pressure interpolation and a second order upwind scheme. The predicted dust concentration distribution showed a good agreement with experimental data adopted from the literature. It was found that the dust concentration distribution was influenced by mean velocity and turbulence flow. The simulation results suggest that the cornstarch concentration inside the silo was always above the lower explosion limit (LEL), hence requiring a mitigating action or a control system to reduce the explosion risk.

LES of incompressible heat and fluid flows past a square cylinder at high Reynolds numbers

This paper presents large eddy simulation (LES) results of incompressible heat and fluid flows around a square cylinder (SC) at zero incident angle at high Reynolds numbers (Re) in the range from 1.25×105 to 3.5×105. LES results are obtained on the basis of swirling strength based sub-grid model, and a higher order upwind scheme developed with respect to the Taylor expansion. It was found that, for the zero incident SC wake flows at a Reynolds number in the range {Re5 = Re/105 ∈ [1.25, 3.5]}, the Strouhal number equals to 0.1079, completely independent of the Reynolds number; the coefficient of drag is around 1.835 with an uncertainty of about 1.9%, almost non-sensitive to the Re. When Re is beyond 3.0×105, the time-averaged peak value of sub-grid viscosity is over 340, implying that the role of sub-grid model is crucial in some regions where vortex motion is active and vortex interaction is intense. The time–spanwise (t-z) averaged sub-grid viscosity ratio profiles and the profiles of fluctuations of the sub-grid viscosity ratio and velocity components at four locations downstream of the SC are presented. The fields of the t-z averaged sub-grid viscosity ratio, and the instantaneous fields of streamwise and spanwise vorticities are also reported and discussed. The predicted mean Nusselt number is compared with empirical correlations, revealing that swirling strength based LES has its potential in predicting natural and industrial flows.

Applying Explicit Schemes to the Korteweg-de Vries Equation

The water wave soliton is a result of a dynamic balance between dispersion and nonlinear effects. It brings together many branches of mathematics, some of which touch on deep ideas. The Korteweg-de Vries equation is typical of all model equations of nonlinear waves in the soliton phenomena. Four explicit difference schemes are used in order to approximate the Korteweg-de Vries equation, namely; (a) a First order upwind scheme, (b) the Zabusky-Kruskal scheme, (c) the Lax-Wendroff scheme, and (d) the Walkley scheme. Our main interest was to analyse which explicit scheme among the four performs well when implemented to the KdV equation to produce the best soliton solution. Hence, reviewing and considering existing schemes. Three sets of initial data are used to explore the numerical approximations. Two, including the data proposed by Zabusky and Kruskal, involve a single soliton wave, whilst the other involves the separation into two solitons, which will interact in time. These initial conditions and periodic boundary conditions are described in detail taking into account physical, mathematical and computational considerations. Accuracy, consistency and Fourier stability in regard to the four explicit schemes for the Korteweg-de Vries equation are discussed. Numerical results are reported for a single soliton solution and the separation into two solitons with different velocities are investigated. Graphical results are presented to show how well these four schemes agree well with each other. After comparing the four explicit schemes, the best scheme was the Zabusky and Kruskal scheme since it is a two-step scheme, which uses the explicit leapfrog finite difference scheme and was good for low amplitudes and less running time was needed than the other three explicit schemes.

Predicting the performance of a high head Francis turbine using a fully implicit mixing plane

In the present paper numerical investigations of a complete high head Francis turbine comprehensive of a spiral casing, stay and guide vanes and draft tube have been performed at three operating conditions, namely at part load (PL), best efficiency point (BEP), and high load (HL). The main target of the investigations is to assess the prediction accuracy of a reduced domain of the complete turbine using a novel mixing-plane formulation.The computational domain is simplified simulating one single passage of the runner, thus assuming rotational periodicity and steady state conditions. The results were compared with experimental data published by the workshop organization.All CFD simulations were performed at model scale with an in-house adapted, 3D, unstructured, object-oriented finite volume code based on the OpenFOAM-V2.2 framework and designed to solve steady-state incompressible RANS-Equations. The pressure-based solver uses a SIMPLE-C like algorithm and is capable of handling multiple references of frame (MRF). The influence of the turbulence has been considered applying the shear-stress transport model (SST). Full second order upwind scheme for advection discretization has been used for all computations.

A numerical study of steady 2D flow around NACA 0015 and NACA 0012 hydrofoil with free surface using VOF method

Accurate simulation of turbulent free surface flows around surface ships has a central role in the optimal design of such naval vessels. The flow problem to be simulated is rich in complexity and poses many modeling challenges because of the existence of breaking waves around the ship hull, and because of the interaction of the two-phase flow with the turbulent boundary layer. In this paper, our goal is to estimate the lift and drag coefficients for NACA 0012 of hydrofoil advancing in calm water under steady conditions with free surface and emerged NACA 0015. The commercial CFD software FLUENT version 14 is used for the computations in the present study. The calculated grid is established using the code computer GAMBIT 2.3.26.The shear stress k-ωSST model is used for turbulence modeling and the volume of fluid technique is employed to simulate the free-surface motion. In this computation, the second order upwind scheme is used for discretizing the convection terms in the momentum transport equations, the Modified HRIC scheme for VOF discretisation. The results obtained compare well with the experimental data.

Detached-eddy Simulation of Subsonic Flow Past a Delta Wing

To alleviate the effect caused by the numerical dissipation of the second order upwind schemes in Detached-Eddy Simulation (DES), an improved second order hybrid scheme is established through modifying the dissipation term of the standard Roe's scheme and the numerical dissipation of the scheme can be self-adapted according to the flow field information. To validate the numerical method, DES of subsonic flow past the 65° swept delta wing is performed at high angle of attack on unstructured/hybrid meshes. Comparison with experimental data shows that the accuracy of the turbulence simulation is improved with the use of the hybrid scheme.

Combined Free and Forced Convection in an Enclosure with different Ventilation Arrangements

The objective of this study is to unveil the effect of different ventilation arrangements in a rectangular enclosure subjected to combined free and forced convection. A heat source is flush mounted at the bottom of the enclosure. The equations of mass, momentum and energy conservation are solved using a control-volume-based technique utilized by Ansys Fluent. A pressure based solver and second order upwind scheme is used for the solution purpose. The non-dimensional variables Richardson number (Ri) and Reynolds number (Re) were varied in the course of parametric study. It is observed that increasing the number of ventilation ports complicates the flow structure inside the enclosure and hence a justified provision of exit openings should be provided. The average heat source surface temperatures in the case of three ventilation ports are higher than the case of two ventilation ports beyond the Ri values of 2.0.

CFD Simulation of Airflow in the Side-Platform Stations

Design of an appropriate ventilation system is one of the main concerns in the construction of subways. In this paper, thermal comforts of side-platform stations are investigated by numerical methods. For the numerical simulation, grids are generated by structured methods and governing equations are discretized by finite volume methods. In the numerical simulation, second order upwind and second order central difference scheme are used to consider the convection and diffusion terms of momentum and energy equations, respectively. Results of this study show that in term of thermal comfort, region next to the hallway and the middle part of the platform is the worst zones of the station. In these zones, air velocity is very low (less than 0.5m/s) and the air temperature reaches to 34°C which is more than its limit (Tmax≤32°C).

Multiphase lattice Boltzmann flux solver for incompressible multiphase flows with large density ratio

A multiphase lattice Boltzmann flux solver (MLBFS) is proposed in this paper for incompressible multiphase flows with low- and large-density-ratios. In the solver, the flow variables at cell centers are given from the solution of macroscopic governing differential equations (Navier–Stokes equations recovered by multiphase lattice Boltzmann (LB) model) by the finite volume method. At each cell interface, the viscous and inviscid fluxes are evaluated simultaneously by local reconstruction of solution for the standard lattice Boltzmann equation (LBE). The forcing terms in the governing equations are directly treated by the finite volume discretization. The phase interfaces are captured by solving the phase-field Cahn–Hilliard equation with a fifth order upwind scheme. Unlike the conventional multiphase LB models, which restrict their applications on uniform grids with fixed time step, the MLBFS has the capability and advantage to simulate multiphase flows on non-uniform grids. The proposed solver is validated by several benchmark problems, such as two-phase co-current flow, Taylor–Couette flow in an annulus, Rayleigh–Taylor instability, and droplet splashing on a thin film at density ratio of 1000 with Reynolds numbers ranging from 20 to 1000. Numerical results show the reliability of the proposed solver for multiphase flows with high density ratio and high Reynolds number.

The effect of thermal radiation on the heat transfer characteristics of lid-driven cavity with a moving surface

Purpose – There are industrial applications for varying speed lid-driven flow and heat transfer such as the float glass process where the glass film stretches or thickens depending on the desired thickness. Hence the tin cavity underneath or the nitrogen cavity above is being driven by a variable speed. The purpose of this paper is to simulate such behavior. Design/methodology/approach – Numerical solution of variable speed lid-driven cavity is carried out with thermal radiation being considered using control volume approach and staggered grid and applying the SIMPLE algorithm. Transient simulation is used for 2D model in the present study. Second order upwind schemes were used for discretization of momentum, energy equations and time. Findings – Under laminar conditions, thermal radiation plays a significant role in the heat transfer characteristics of the lid-driven cavity. This effect is more significant for blackbody radiation and decreases as the surface emissivity decreases. Nusselt number (Nu) behavior lies between these two limiting case profiles considering constant speed profiles of both maximum and minimum lid velocities, respectively. In addition, local Nu values at the tip where higher than those at the top of the cavity that is stagnant. Research limitations/implications – The study is limited to laminar flow case. Practical implications – The applications of this study can be found in float glass process where the glass film stretches or thickens depending on the desired thickness. Hence the tin cavity underneath or the nitrogen cavity above is being driven by a variable speed. Another application involves casting of plastic films. The molten polymer leaves the die with a considerable thickness and high temperature. The film is then trenched to reach its final thickness. In this case, usually there is no actual cavity above or below the film but one can approximate the problem as such. Other similar applications do exist in food drying and processing where the conveyer belt is in portions and their speed may not be the same in different section of the processing oven. Originality/value – To the best of the authors knowledge, no study in the literature addressed the effect of thermal radiation in lid-driven cavities with variable speed

Numerical simulation of free surface water wave for the flow around NACA 0015 hydrofoil using the volume of fluid (VOF) method

The surface wave generated by flow around NACA 0015 hydrofoil moving near free surface of water is simulated numerically in this study. The two-dimensional implicit finite volume method (FVM) is applied to solve Reynolds Averaged Navier–Stokes (RANS) equation. The realizable κ-ε turbulence model has been implemented to capture turbulent flow around the hydrofoil in the free surface zone at different submergence ratios (h/c). The volume of fluid (VOF) method has been used to determine the free surface effect of water. For pressure–velocity coupling, SIMPLEC (Semi IMPlicit Linked Equations Consistent) algorithm is employed. The second order upwind scheme is applied for discretization of momentum, volume fraction, turbulent kinetic energy and turbulent dissipation rate. At first NACA 0012 hydrofoil section is analyzed at h/c=0.91 and the result is validated by comparing with the published experimental result. Finally, the analysis is carried out with NACA 0015 hydrofoil section for seven h/c ratios, ranging from 0.91 to 4.0. The profile of the waves, the contours of velocity magnitude and static pressure near the hydrofoil and free surface, and the values of lift and drag coefficients are computed at Fn=0.5711, Re=1.592×105 for those submergence ratios.