Accurate Upwind Scheme Research Articles

Flux-splitting dispersion-relation-preserving dual-compact upwind scheme for solving the Maxwell’s equations on non-staggered grids

This study proposes a flux-splitting Maxwell’s equations solver for modeling electromagnetic waves in two-dimensional non-staggered grids. A fifth-order spatially accurate dual-compact upwind scheme was developed in a three-point grid stencil to approximate the first-order derivative term. The integrity of the proposed finite-difference time-domain method for solving TM-mode Maxwell’s equations verified using two-dimensional test problems. The benchmark Mie-scattering problem was also shown to be in good agreement with the semi-analytic result.

A study of convective flux schemes for aerospace flows

This paper presents the effects of some convective flux computation schemes on boundary layer and shocked flow solutions. Second-order accurate centered and upwind convective flux computation schemes are discussed. The centered Jameson scheme, plus explicitly added artificial dissipation terms are considered. Three artificial dissipation models, namely a scalar and a matrix version of a switched model, and the CUSP scheme are available. Some implementation options regarding these methods are proposed and addressed in the paper. For the upwind option, the Roe flux-difference splitting scheme is used. The CUSP and Roe schemes require property reconstructions to achieve second-order accuracy in space. A multidimensional limited MUSCL interpolation method is used to perform property reconstruction. Extended multidimensional limiter formulation and implementation are here proposed and verified. Theoretical flow solutions are used in order to provide a representative testbed for the current study. It is observed that explicitly added artificial dissipation terms of the centered scheme may nonphysically modify the numerical solution, whereas upwind schemes seem to better represent the flow structure.

An efficient higher order compact scheme to capture heat transfer solutions in spherical geometry

A higher order compact scheme (HOCS) is developed for solving energy equation in spherical polar coordinates for the flow past a sphere. The velocity components are obtained by solving the Navier–Stokes equations using HOCS in spherical geometry. The convection terms in the momentum and energy equations are handled in an effective manner so as to get fourth order accurate solutions. The higher order compact scheme is made efficient by combining with the coarse grid correction based multigrid method. In the absence of an exact solution, the order of accuracy of the results is illustrated. The results are compared with first order accurate upwind scheme, second order accurate defect correction technique(DC) and other recent numerical results of the literature. It is found that HOCS captures heat transfer solutions accurately when compared to other conventional schemes at higher values of Re and Pr. The angular variation of thermal boundary layer thickness is also presented.

Upwind schemes for the wave equation in second-order form

We develop new high-order accurate upwind schemes for the wave equation in second-order form. These schemes are developed directly for the equations in second-order form, as opposed to transforming the equations to a first-order hyperbolic system. The schemes are based on the solution to a local Riemann-type problem that uses d’Alembert’s exact solution. We construct conservative finite difference approximations, although finite volume approximations are also possible. High-order accuracy is obtained using a space-time procedure which requires only two discrete time levels. The advantages of our approach include efficiency in both memory and speed together with accuracy and robustness. The stability and accuracy of the approximations in one and two space dimensions are studied through normal-mode analysis. The form of the dissipation and dispersion introduced by the schemes is elucidated from the modified equations. Upwind schemes are implemented and verified in one dimension for approximations up to sixth-order accuracy, and in two dimensions for approximations up to fourth-order accuracy. Numerical computations demonstrate the attractive properties of the approach for solutions with varying degrees of smoothness.

On the development of a high-order compact scheme for exhibiting the switching and dissipative solution natures in the Camassa–Holm equation

In this paper a two-step iterative solution algorithm for solving the Camassa–Holm equation, which involves only the first-order derivative term, is presented. In each set of the u − P and u − m differential equations, one is governed by the inviscid nonlinear convection–reaction equation for the time-evolving fluid velocity component along the horizontal direction. The other equation is known as the inhomogeneous Helmholtz equation. The resulting reduction of differential order facilitates us to develop the flux discretization scheme in a stencil with comparatively fewer points. For accurately predicting the unidirectional propagation of the shallow water wave, the modified equation analysis for eliminating several leading discretization error terms and the Fourier analysis for minimizing a particular type of wave-like error are employed. In this study, the fifth-order spatially accurate combined compact upwind scheme is developed in a three-point stencil for approximating the first-order derivative term. For the purpose of retaining a long-term accurate Hamiltonian and multi-symplectic geometric structures in Camassa–Holm equation, the time integrator (or time-stepping scheme) chosen in this study should conserve symplecticity. Another main emphasis of conducting the present calculation of Camassa–Holm equation is to shed light on the conservation of Hamiltonians up to the time before wave breaking. We also intended to elucidate the switching scenario by virtue of the peakon–peakon interaction problem and the dissipative scenario after the time of head-on collision in the peakon–antipeakon interaction problem.

A Multigrid-Accelerated Code on Graded Cartesian Meshes for 2D Time-Dependent Incompressible Viscous Flows

The present work deals with the development of a multigrid-assisted solver for the 2D time-dependent incompressible Navier-Stokes equations on graded Cartesian meshes. As finite difference method is used to discretize the governing equations on nonuniform staggered grids, a transformation of the governing equations from the physical space to the computational space is performed. To obtain second-order time accuracy a fractional-step method is employed. The convective terms are discretized using a third-order accurate upwind scheme and the viscous terms are centrally differenced to fourth-order accuracy. To improve the time-wise efficiency of the code a multigrid technique is employed to solve the pressure-Poisson equation that is required to be solved at every time-step. To establish the accuracy and performance of the code the standard 2D lid-driven cavity flow is computed for unsteady, periodic and asymptotically obtained steady solution for a wide range of Reynolds numbers (Re). The code is then used to compute the transient and asymptotically approached steady flows in a hitherto unexamined problem of two-sided lid-driven square cavity, which involves gradual development of a free shear layer and accompanying off-corner vortices. The computations also show that for this configuration, at Re=2000, there exists a steady solution, about which there was some doubt earlier. The reliability of all known and unknown results in the paper is carefully established and efficiency of the method in respect of grid economy is demonstrated.

Assessment of Large-Eddy Simulation of Internal Separated Flow

This paper presents a systematic numerical investigation of different implicit large-eddy simulations (LESs) for massively separated flows. Three numerical schemes, a third-order accurate monotonic upwind scheme for scalar conservation laws (MUSCL) scheme, a fifth-order accurate MUSCL scheme, and a ninth-order accurate weighted essentially non-oscillatory (WENO) method, are tested in the context of separation from a gently curved surface. The case considered here is a simple wall-bounded flow that consists of a channel with a hill-type curvature on the lower wall. The separation and reattachment locations, velocity, and Reynolds stress profiles are presented and compared against solutions from classical LES simulations.

Insights on a sign‐preserving numerical method for the advection–diffusion equation

AbstractIn this paper we explore theoretically and numerically the application of the advection transport algorithm introduced by Smolarkiewicz to the one‐dimensional unsteady advection–diffusion equation. The scheme consists of a sequence of upwind iterations, where the initial iteration is the first‐order accurate upwind scheme, while the subsequent iterations are designed to compensate for the truncation error of the preceding step. Two versions of the method are discussed. One, the classical version of the method, regards the second‐order terms of the truncation error and the other considers additionally the third‐order terms. Stability and convergence are discussed and the theoretical considerations are illustrated through numerical tests. The numerical tests will also indicate in which situations are advantageous to use the numerical methods presented. Copyright © 2008 John Wiley & Sons, Ltd.

NUMERICAL SCHEME OF THE WAHA CODE

This paper describes the numerical scheme used in the WAHA code that was developed within the WAHALoads project for simulations of fast transients in 1D piping systems. Twofluid model equations are solved with an operator splitting procedure. A second-order accurate non-conservative characteristic upwind scheme is used to solve the hyperbolic part of the equations with the non-relaxation source terms, while the relaxation source terms are treated in the second step of the operator splitting procedure. The water-vapour properties are calculated with a newly developed set of subroutines that use pre-tabulated water properties. Special models that were developed for the treatment of the abrupt area-changes and branches in the piping systems are described. Various test cases, which were used to test the accuracy of the basic numerical scheme and the accompanying numerical models, are described together with typical results of the simulations.

Mixed convection from an isolated spherical particle

A numerical study on mixed convection around a hot spherical particle moving vertically downwards in a still fluid medium has been made. The flow field is considered to be axisymmetric for the range of Reynolds number (based on the diameter and the settling velocity of the particle) considered. A third-order accurate upwind scheme is employed to compute the flow field and the temperature distribution. The form of the wake and the thermal field is analyzed for several values of Grashof number and the Reynolds number. The influence of buoyancy on drag and the rate of heat transfer are studied. At moderate Reynolds number, recirculating eddy develops in the downstream of the sphere. With the rise of surface temperature this eddy collapses and the fluid adjacent to the heated surface develops into a buoyant plume above the sphere. The increase in surface temperature of the sphere delays the flow separation. Our results show that the drag force and the rate of heat transfer strongly depend on Grashof number for the moderate values of Reynolds number. The conjugate heat transfer from the moving sphere is also addressed in the present paper. We have compared our computed solution with several empirical and asymptotic expressions available in the literature and found them in good agreement.

Laminar Mixed Convection in a Parallel Two-Sided Lid-Driven Differentially Heated Square Cavity Filled with a Fluid-Saturated Porous Medium

Two-dimensional, steady, mixed convection flow in an enclosure filled with a Darcian fluid-saturated uniform porous medium is considered. Both the left wall and the right wall are moving upwards with same velocity. They are maintained at cold and hot constant temperatures respectively. The top and the bottom walls are fixed and are thermally insulated. The governing equations are normalized and solved numerically with boundary conditions by finite volume approach using third order accurate upwinding scheme (deferred QUICK). To validate the code, comparisons are made with previously published work and are found to be in excellent agreement. The study is conducted by varying the key parameters i.e., Richardson number (Ri = Gr/Re 2), Darcy number (Da = κ/H 2) and Grashof number (Gr = g β H 3ΔT/ν 2) and fixing Prandtl number (Pr = 0.71). A parametric study is conducted and a set of streamline and isotherm plots are presented. The average Nusselt number reaches a value of 1 asymptotically when the Ri is gradually increased for Gr up to 103. For Gr = 104, the asymptotic value is 1.5. An empirical heat transfer correlation is also presented.

Progress in the CFD Modeling of Flow Instabilities in Anatomical Total Cavopulmonary Connections

Intrinsic flow instability has recently been reported in the blood flow pathways of the surgically created total-cavopulmonary connection. Besides its contribution to the hydrodynamic power loss and hepatic blood mixing, this flow unsteadiness causes enormous challenges in its computational fluid dynamics (CFD) modeling. This paper investigates the applicability of hybrid unstructured meshing and solver options of a commercially available CFD package (FLUENT, ANSYS Inc., NH) to model such complex flows. Two patient-specific anatomies with radically different transient flow dynamics are studied both numerically and experimentally (via unsteady particle image velocimetry and flow visualization). A new unstructured hybrid mesh layout consisting of an internal core of hexahedral elements surrounded by transition layers of tetrahedral elements is employed to mesh the flow domain. The numerical simulations are carried out using the parallelized second-order accurate upwind scheme of FLUENT. The numerical validation is conducted in two stages: first, by comparing the overall flow structures and velocity magnitudes of the numerical and experimental flow fields, and then by comparing the spectral content at different points in the connection. The numerical approach showed good quantitative agreement with experiment, and total simulation time was well within a clinically relevant time-scale of our surgical planning application. It also further establishes the ability to conduct accurate numerical simulations using hybrid unstructured meshes, a format that is attractive if one ever wants to pursue automated flow analysis in a large number of complex (patient-specific) geometries.

A new conservative unsplit method for the solution of the Vlasov equation

We have developed a new conservative method for solving the Vlasov equation without using any splitting technique. Our goal is to maintain the positivity of the distribution function and to avoid un-physical oscillations which might lead to numerical instabilities. Based on a finite volume conservative discretization of the conservative form of the Vlasov equation we implemented a highly accurate second-order upwind scheme. In order to avoid un-physical oscillations and their possible numerical instability we apply a flux-limiter in the second order. We validate our new Vlasov solver by considering standard cases of one-dimensional current-driven ion-acoustic instabilities solving a Vlasov–Ampère set of equations.

Toward Excellence in Turbomachinery Computational Fluid Dynamics: A Hybrid Structured-Unstructured Reynolds-Averaged Navier-Stokes Solver

Abstract A three-dimensional hybrid structured-unstructured Reynolds-averaged Navier-Stokes (RANS) solver has been developed to simulate flows in complex turbomachinery geometries. It is built by coupling an existing structured computational fluid dynamics (CFD) solver with a newly developed unstructured-grid module via a conservative hybrid-grid interfacing algorithm, so that it can get benefits from the both structured and unstructured grids. The unstructured-grid module has been developed with consistent numerical algorithms, data structure, user interface and parallelization to those of the structured one. The numerical features of the hybrid RANS solver are its second-order accurate upwind scheme in space, its SGS implicit formulation of time integration, and its accurate modeling of steady/unsteady boundary conditions for multistage turbomachinery flows. The hybrid-grid interfacing algorithm is essentially an extension of the conservative zonal approach that has been previously applied on the mismatched zonal interface of the structured grids, and it is fully conservative and also second-order accurate. Due to the mismatched grids allowed at the block interface, users would have great flexibility to build the hybrid grids even with different structured and unstructured grid generators. The performance of the hybrid RANS solver is assessed with a variety of validation and application examples, through which the hybrid RANS solver has been demonstrated to be able to cope with the flows in complex turbomachinery geometries and to be promising for the future industrial applications.

Implicit schemes with large time step for non‐linear equations: application to river flow hydraulics

AbstractIn this work, first‐order upwind implicit schemes are considered. The traditional tridiagonal scheme is rewritten as a sum of two bidiagonal schemes in order to produce a simpler method better suited for unsteady transcritical flows. On the other hand, the origin of the instabilities associated to the use of upwind implicit methods for shock propagations is identified and a new stability condition for non‐linear problems is proposed. This modification produces a robust, simple and accurate upwind semi‐explicit scheme suitable for discontinuous flows with high Courant–Friedrichs–Lewy (CFL) numbers.The discretization at the boundaries is based on the condition of global mass conservation thus enabling a fully conservative solution for all kind of boundary conditions.The performance of the proposed technique will be shown in the solution of the inviscid Burgers' equation, in an ideal dambreak test case, in some steady open channel flow test cases with analytical solution and in a realistic flood routing problem, where stable and accurate solutions will be presented using CFL values up to 100. Copyright © 2004 John Wiley & Sons, Ltd.

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.

Analysis of aerodynamic noise generated from inclined circular cylinder

Making clear the generation mechanism of fluid dynamic noise is essential to reduce noise deriving from turbomachinery. The analysis of the aerodynamic noise generated from circular cylinder is carried out numerically and experimentally in a low noise wind tunnel. In this study, aerodynamic sound radiated from a circular cylinder in uniform flow is predicted numerically by the following two step method. First, the three-dimensional unsteady incompressible Navier-Stokes equation is solved using the high order accurate upwind scheme. Next, the sound pressure level at the observed point is calculated from the fluctuating surface pressure on the cylinder, based on modified Lighthill-Curl’s equation. It is worth to note that the noise generated from the model is reduced rapidly when it is inclined against the mean flow. In other words, the peak level of the radiated noise decreases rapidly with inclination of the circular cylinder. The simulated SPL for the inclined circular cylinder is compared with the measured value, and good agreement is obtained for the peak spectrum frequency of the sound pressure level and tendency of noise reduction. So we expect that the change of flow structures makes reduction of the aerodynamic noise from the inclined models.

An implicit unfactored method for unsteady turbulent compressible flows with moving boundaries

An implicit, unfactored solver for the simulation of unsteady turbulent flows around moving solid bodies is presented. The method is applied to solve the turbulence transport equations in conjunction with the Navier–Stokes equations in a strong coupled fashion. A third-order accurate upwind scheme is used for discretising the inviscid fluxes and Newton-type sub-iterations are employed to obtain time-accurate solutions on a moving mesh. The flow around an oscillating aerofoil is computed as a test problem and calculations are presented both for laminar and turbulent flows. Initially, laminar flow calculations over an oscillating NACA 0012 aerofoil at various Mach numbers were carried out in order to assess the accuracy of the method in the prediction of the dynamic-stall vortex under laminar flow conditions. Furthermore, calculations were performed for quasi-steady and fully unsteady turbulent flows over the NACA 0015 aerofoil using various turbulence models at different flow conditions corresponding to attached unsteady flow as well as light- and deep-stall cases. It was found that the numerical results for unsteady, turbulent flow calculations over an aerofoil depend strongly on the turbulence model used.

High-order upwind method for hyperbolic grid generation

A new hyperbolic grid generation method based on a high-order accurate upwind approximation is presented and developed for two- and three-dimensional spaces in this paper. The method uses a system of hyperbolic equations for grid generation. Due to a hyperbolic property of the grid generation equations, there exist characteristic directions and high-order TVD upwind scheme can be applied to the directions. The method employs a three-point backward difference scheme for descritization of a derivative in a marching direction, while a high-order accurate upwind scheme using MUSCL interpolation with minmod limiter is employed according to the plus or minus sign of eigenvalues of coefficient matrices to descritize derivatives in the other directions. Making use of the upwind scheme the present hyperbolic grid generation method needs no artificial damping term, which a conventional hyperbolic grid generation method requires to prevent oscillations. The method incorporates a subiteration procedure to solve nonlinear equations at each marching step in order to obtain both sufficient robustness and satisfactory grid orthogonality realized by the high-order upwind scheme. For two dimensions, the Newton subiteration is performed at each marching step before advancing to next grid line. For three dimensions, a new pseudo-time subiteration approach is introduced and incorporated in the method to achieve further robustness. It is demonstrated that in the hyperbolic grid generation the first-order accurate upwind scheme is too dissipative to generate an orthogonal grid, although the higher-order accurate upwind scheme can generate the orthogonal grid. The generated grids around bodies of complex geometry with sharp edges, deep concave and high convex show excellent orthogonality and smoothness, and demonstrate promising properties of the method.

A numerical analysis of vortex shedding within a confined channel flow

The numerical analysis of two-dimensional laminar vortex shedding from a rectangular cylinder within a confined channel flow is presented. This study, carried out using a computational fluid dynamics (CFD) code based on the SIMPLEST algorithm, considers the influence of numerical diffusion on the prediction of the vortex shedding frequency. The computational analysis compares the commonly used first-order accurate UPWIND scheme with the well-known third-order scheme QUICK and its derivative SMART, used for the discretization of convective transport. For the temporal differencing, a fully implicit scheme has been adopted. Plots of Strouhal number against Reynolds number suggest that the implementation of a higher-order scheme is beneficial for the accurate capture of the vortex shedding transient in unsteady flows of this nature.