High-order Upwind Scheme Research Articles

Rotor wake capture improvement based on high-order spatially accurate schemes and chimera grids

A high-order upwind scheme has been developed to capture the vortex wake of a helicopter rotor in the hover based on chimera grids. In this paper, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate the higher-order left and right states across a cell interface with the Roe Riemann solver updating inviscid flux, and is compared with the monotone upwind scheme for scalar conservation laws (MUSCL). For profitably capturing the wake and enforcing the period boundary condition, the computation regions of flows are discretized by using the structured chimera grids composed of a fine rotor grid and a cylindrical background grid. In the background grid, the mesh cells located in the wake regions are refined after the solution reaches the approximate convergence. Considering the interpolation characteristic of the WENO scheme, three layers of the hole boundary and the interpolation boundary are searched. The performance of the schemes is investigated in a transonic flow and a subsonic flow around the hovering rotor. The results reveal that the present approach has great capabilities in capturing the vortex wake with high resolution, and the WENO scheme has much lower numerical dissipation in comparison with the MUSCL scheme.

High-order upwind methods based on C2-continuous two-node integrated-RBF elements for viscous flows

In this paper, 2-node integrated radial basis function elements (IRBFEs)[CMES, vol.72, no.4, pp.299-334, 2011] are further developed for the simulation of incompressible viscous flows in two dimensions. Emphasis is placed on (i) the incorporation of C2-continuous 2-node IRBFEs into the subregion and point collocation frameworks for the discretisation of the stream function-vorticity formulation on Cartesian grids; and (ii) the development of high order upwind schemes based on 2-node IRBFEs for the case of convection-dominant flows. High levels of accuracy and efficiency of the present methods are demonstrated by solutions of several benchmark problems defined on rectangular and non-rectangular domains.

A stable and accurate control-volume technique based on integrated radial basis function networks for fluid-flow problems

Radial basis function networks (RBFNs) have been widely used in solving partial differential equations as they are able to provide fast convergence. Integrated RBFNs have the ability to avoid the problem of reduced convergence-rate caused by differentiation. This paper is concerned with the use of integrated RBFNs in the context of control-volume discretisations for the simulation of fluid-flow problems. Special attention is given to (i) the development of a stable high-order upwind scheme for the convection term and (ii) the development of a local high-order approximation scheme for the diffusion term. Benchmark problems including the lid-driven triangular-cavity flow are employed to validate the present technique. Accurate results at high values of the Reynolds number are obtained using relatively-coarse grids.

High effective WENO scheme for transonic inviscid flow computation on a helicopter rotor in hover

A high-order upwind scheme with high efficiency has been developed to compute the inviscid flow of a helicopter rotor in hover. For rotary-wing wake capturing, an improved fifth-order weighted essentially non-oscillatory (WENO) scheme is adopted to interpolate higher order left and right states across a cell interface with Roe Riemann solver updating inviscid flux. To improve the efficiency and convergence to steady state, three-level V-cycle multigrid relaxation scheme is used. The performance of the schemes is investigated in a transonic inviscid flow around hovering rotor. The results reveal that WENO has the great capabilities to capture shock with high resolution and has a lower numerical dissipation than MUSCL for capturing wake vorticity. Moreover, multigrid scheme can accelerate convergence to a great extent.

Hybrid weighted essentially non-oscillatory schemes with different indicators

A key idea in finite difference weighted essentially non-oscillatory (WENO) schemes is a combination of lower order fluxes to obtain a higher order approximation. The choice of the weight to each candidate stencil, which is a nonlinear function of the grid values, is crucial to the success of WENO schemes. For the system case, WENO schemes are based on local characteristic decompositions and flux splitting to avoid spurious oscillation. But the cost of computation of nonlinear weights and local characteristic decompositions is very high. In this paper, we investigate hybrid schemes of WENO schemes with high order up-wind linear schemes using different discontinuity indicators and explore the possibility in avoiding the local characteristic decompositions and the nonlinear weights for part of the procedure, hence reducing the cost but still maintaining non-oscillatory properties for problems with strong shocks. The idea is to identify discontinuity by an discontinuity indicator, then reconstruct numerical flux by WENO approximation in discontinuous regions and up-wind linear approximation in smooth regions. These indicators are mainly based on the troubled-cell indicators for discontinuous Galerkin (DG) method which are listed in the paper by Qiu and Shu (J. Qiu, C.-W. Shu, A comparison of troubled-cell indicators for Runge–Kutta discontinuous Galerkin methods using weighted essentially non-oscillatory limiters, SIAM Journal of Scientific Computing 27 (2005) 995–1013). The emphasis of the paper is on comparison of the performance of hybrid scheme using different indicators, with an objective of obtaining efficient and reliable indicators to obtain better performance of hybrid scheme to save computational cost. Detail numerical studies in one- and two-dimensional cases are performed, addressing the issues of efficiency (less CPU time and more accurate numerical solution), non-oscillatory property.

An upwind control-volume method based on integrated RBFs for fluid-flow problems

This paper is concerned with the development of a high-order upwind conservative numerical method for the simulation of flows of a Newtonian fluid. The domain is discretised using a Cartesian grid from which non-overlapping rectangular control-volumes are formed. Line integrals arising from the integration of the diffusion and convection terms over control volumes are evaluated using the middle-point rule. The diffusion term is approximated using two different schemes, namely one-dimensional integrated radial basis function networks (1D-IRBFN) and central differences. The convection term is effectively treated by a high-order upwind scheme incorporating 1D-IRBFNs with the deferred-correction strategy. Several test problems governed by the Burgers and the Navier-Stokes equations are simulated, which show that the proposed technique is stable, accurate and converges well.

Nearly All-Speed, Stabilized Time-Accurate Upwind Scheme on Unstructured Grid

time-accurate, upwind, finite volume method for computing compressible flows on unstructured grids is presented. The method is second-order-accurate in space and time and yields high resolution in the presence of discontinuities. In the basic Euler and Navier-Stokes upwind scheme, many concepts of high-order upwind schemes are adopted: the surface flux integrals are carefully treated, a Cauchy-Kowalewski time-stepping scheme is used in the time-marching stage, and a multidimensional limiter is applied in the reconstruction stage. However, even with these up-to-date improvements, the basic upwind scheme is still plagued by the so-called pathological behaviors (for example, the carbuncle, the expansion shock, etc.), which are mostly triggered due to some undesirable local numerical instability. simple multidimensional dissipation model is used to systematically suppress such behaviors and stabilize the scheme for flows at high Mach numbers, whereas for flows at very low Mach number (for example, M = 0.02), it is found that computation can be directly carried out without invoking preconditioning. The modified, stabilized scheme is referred to as the enhanced time-accurate upwind scheme (Loh, C. Y., and Jorgenson, P. C. E., A Time Accurate Upwind Unstructured Finite Volume Method for Compressible Flow with Cure of Pathological Behaviors, AIAA Paper 2007-4463, 2007.) in this paper. The unstructured grid capability renders flexibility for use in complex geometry, and the present enhanced time-accurate upwind Euler and Navier-Stokes scheme is capable of handling a broad spectrum of flow regimes from high supersonic to subsonic at very low Mach number, appropriate for both computational fluid dynamics and computational aeroacoustics. Numerous examples are included to demonstrate the robustness of the scheme.

Compact upwind schemes on adaptive octrees

Compact high-order upwind schemes using reconstruction from cell-averages are derived for application with the compressible three-dimensional Navier–Stokes equations. An adaptive-octree mesh, combined with the Adams–Bashforth–Moulton family of predictor–corrector schemes, provides a conservative high-order time-integration platform supporting localized h-refinement and timestep sub-cycling. Numerical examples for smooth flows demonstrate the improvement over explicit upwind schemes and formal accuracy of the schemes, as well as the behavior in wall-bounded regions, and the resolution of a broad wavenumber spectrum.

A localized collocation meshless method (LCMM) for incompressible flows CFD modeling with applications to transient hemodynamics

The current paper reports on the development and validation of a localized collocation meshless method (LCMM) to model laminar incompressible flows. A high order upwinding scheme was devised to dampen the numerical oscillations arising in convection-dominated flows. Subsequently, the LCMM was analytically validated and demonstrated to yield third-order accurate solutions when compared to a benchmark analytical decaying vortex solution. Numerical validations are provided by comparison with the finite volume commercial (FVM) solver Fluent 6.2. The flow geometry for the numerical validation arises from a biomedical application that consists of modeling blood flow in the inter-connection between a bypass graft and an artery. Very good agreement was found between the LCMM and the FVM.

High-resolution compact upwind finite difference methods for linear wave phenomena

A new family of high-resolution compact upwind finite difference schemes has been recently developed by the author for resolving discontinuities in the solution of linear convection problems. The improved method are based on fairly different approaches. By the modified equation analysis, all free parameters have been rigorously determined for obtaining physically more accurate solutions. In addition, by means of the characteristic analysis of the operators, the group velocity of wave packets has been controlled to avoid spurious oscillations while keeping the formal accuracy of the scheme. As for other classical methods, the application of the flux-corrected transport technique has been examined to eliminate or reduce the appearance of over- and under-shoots in discrete solutions. Finally, we have build a class of adaptive compact algorithms based on the non-linear combination of low- and high-order upwind schemes. We present a wide series of experiments for which well-established analytical and numerical results are available. The proposed numerical algorithms are compared with one another and with some conventional and shock-capturing schemes and very good agreement is achieved in computer calculations. The quality of the results is very encouraging and demonstrated the potential of the methods. The numerical experiments are also consistent with the expectations resulting from the theoretical analysis.

High order upwind scheme for modelling turbulent shallow water flow in hydraulic structures

AbstractAn unstructured finite volume model for quasi‐2D free surface flow with wet‐dry fronts and turbulence modelling is presented. The convective flux is discretised with either a an hybrid second‐order/first‐order scheme, or a fully second order scheme, both of them upwind Godunov's schemes based on Roe's average. The hybrid scheme uses a second order discretisation for the two unit discharge components, whilst keeping a first order discretisation for the water depth [2]. In such a way the numerical diffusion is much reduced, without a significant reduction on the numerical stability of the scheme, obtaining in such a way accurate and stable results. It is important to keep the numerical diffusion to a minimum level without loss of numerical stability, specially when modelling turbulent flows, because the numerical diffusion may interfere with the real turbulent diffusion.In order to avoid spurious oscillations of the free surface when the bathymetry is irregular, an upwind discretisation of the bed slope source term [4] with second order corrections is used [2]. In this way a fully second order scheme which gives an exact balance between convective flux and bed slope in the hydrostatic case is obtained. The k – ε equations are solved with either an hybrid or a second order scheme.In all the numerical simulations the importance of using a second order upwind spatial discretisation has been checked [1]. A first order scheme may give rather good predictions for the water depth, but it introduces too much numerical diffusion and therefore, it excessively smooths the velocity profiles. This is specially important when comparing different turbulence models, since the numerical diffusion introduced by a first order upwind scheme may be of the same order of magnitude as the turbulent diffusion. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

False diffusion in numerical simulation of combustion processes in tangential-fired furnace

Numerical simulation serves as one of the most important tools for analyzing coal combustion in Tangentially Fired Furnaces (TFF) with NUMERICAL FALSE DIFFUSION as one key problem that degrades the simulation accuracy, especially for complex flow patterns. False diffusion often completely compromises the accuracy, leading to erroneous predictions. This paper reviews various methods to reduce the numerical diffusion. In computational fluid dynamics (CFD), false diffusion originates from a truncation error of the Taylor series approximation of the derivative and multidimensional discretization effects. Higher-order upwind convective schemes were designed to reduce truncation errors, while grid line adjusting methods were developed to reduce crossflow diffusion. This paper compares numerical and experimental results for isothermal flows to evaluate these methods. Results with the standard upwind scheme in a rectangular Cartesian mesh are compared with results in body-fitted meshes for comprehensive combustion processes in a TFF. Analysis of the false diffusion effect in the x, y, z directions and the artificial viscosity distribution in a rectangular mesh shows where the false diffusion overtakes the real physical diffusion and where the mesh must be refined or grid line must be adjusted to improve TFF combustion simulations.

Heatline method for the visualization of natural convection in a complicated cavity

In this paper, natural convection inside a two-dimensional cavity with a wavy right vertical wall has been carried out. The bottom wall is heated by a spatially varying temperature and other three walls are kept at constant lower temperature. The integral forms of the governing equations are solved numerically using finite-volume method in non-orthogonal body-fitted coordinate system. SIMPLE algorithm with higher-order upwinding scheme are used. The method of numerical visualization of heat transport for convective heat transfer by heatlines is studied. The heatfunction equation in the transformed plane is solved in terms of dimensionless variables. Results are presented in the form of streamlines, isotherms, heatlines, local and average Nusselt number distribution for a selected range of Rayleigh number (10 0–10 6). The results are presented for three different undulations (1–3) with different wave amplitude (0.00–0.10) and a fluid having Prandtl number 0.71.

A variable relaxation scheme for multiphase, multicomponent flow

We propose a variable relaxation scheme for the simulation of 1D, two-phase, multicomponent flow in porous media. For these strongly nonlinear systems, traditional high order upwind schemes are impractical: Riemann solutions are not directly available when the phase behavior is complex, and the systems are weakly hyperbolic at isolated points. Relaxation schemes avoid the dependency on the eigenstructure and nonlinear Riemann solvers by approximating the original system with a strongly hyperbolic linear system. We exploit the known information about the eigenvalues to construct first order and second order variable relaxation schemes with much reduced numerical diffusion as compared to the standard relaxation formulations. The proposed second order variable relaxation scheme is competitive in accuracy and efficiency with a third order component-wise ENO reconstruction, and performs at least as well as second order component-wise TVD schemes.

Evaluation of a bounded high order upwind scheme for 3D incompressible free surface flow computations

In the context of normalized variable formulation (NVF) of Leonard and total variation diminishing (TVD) constraints of Harten, this paper presents an extension of a previous work by the authors for solving unsteady incompressible flow problems. The main contributions of the paper are threefold. First, it presents the results of the development and implementation of a bounded high order upwind adaptative QUICKEST scheme in the 3D robust code (Freeflow), for the numerical solution of the full incompressible Navier–Stokes equations. Second, it reports numerical simulation results for 1D shock tube problem, 2D impinging jet and 2D/3D broken dam flows. Furthermore, these results are compared with existing analytical and experimental data. And third, it presents the application of the numerical method for solving 3D free surface flow problems.

Hopf bifurcation in mixed convection flow inside a rectangular cavity

In the present paper, a study of mixed convection inside a rectangular cavity has been carried out. The Reynolds number ( Re) has been kept at 100 while the Grashof number ( Gr) has been varied between 0, ± 10 4 , ± 10 6 and aspect ratio (AR) (height/width) = 0.5, 1 and 2 keeping the Prandtl number ( Pr) = 1. The two vertical walls are maintained at cold temperature T = 0 . In one case the top-moving wall is maintained at hot T = 1 and the bottom is cold T = 0 and in the other case, the top is cold T = 0 and the bottom is hot T = 1 . The integral form of the governing equations are solved numerically using finite-volume method. SIMPLE algorithm with higher-order upwinding scheme is used. Results are presented in the form of local and average Nusselt number distribution for the range of Grashof number and aspect ratio. The streamlines and isothermal lines are also presented. A Hopf bifurcation has been observed at Gr = - 10 5 for aspect ratio 2. A periodic oscillation of the total kinetic energy (TKE) occurs with the period 4.368 non-dimensional time.

Incompressible Turbulent Flow Simulation Using theκ-ɛModel and Upwind Schemes

In the computation of turbulent flows via turbulence modeling, the treatment of the convective terms is a key issue. In the present work, we present a numerical technique for simulating two-dimensional incompressible turbulent flows. In particular, the performance of the high Reynoldsκ-ɛmodel and a new high-order upwind scheme (adaptative QUICKEST by Kaibara et al. (2005)) is assessed for 2D confined and free-surface incompressible turbulent flows. The model equations are solved with the fractional-step projection method in primitive variables. Solutions are obtained by using an adaptation of the front tracking GENSMAC (Tomé and McKee (1994)) methodology for calculating fluid flows at high Reynolds numbers. The calculations are performed by using the 2D version of theFreeflowsimulation system (Castello et al. (2000)). A specific way of implementing wall functions is also tested and assessed. The numerical procedure is tested by solving three fluid flow problems, namely, turbulent flow over a backward-facing step, turbulent boundary layer over a flat plate under zero-pressure gradients, and a turbulent free jet impinging onto a flat surface. The numerical method is then applied to solve the flow of a horizontal jet penetrating a quiescent fluid from an entry port beneath the free surface.

Derivation of high-order advection–diffusion schemes

Using the interpolation polynomial method, major upwind explicit advection–diffusion schemes of up to fifth-order accuracy are rederived and their properties are explored. The trend emerges that the higher the order of accuracy of an advection scheme, the easier is the task of scheme stabilization and wiggling suppression. Thus, for a certain range of the turbulent diffusion coefficient, the stability interval of third- and fifth-order up-upwind explicit schemes can be extended up to three units of the Courant number (0≤c≤3). Having good phase behavior, advection odd-order schemes are stable within a single computational cell (0≤c≤1). By contrast, even-order schemes are stable within two consecutive grid-cells (0≤c≤2), but exhibit poor dispersive properties. Stemming from the finding that considered higher-order upwind schemes (even, in particular) can be expressed as a linear combination of two lower-order ones (odd in this case), the best qualities of odd- and even-order algorithms can be blended within mixed-order accuracy schemes. To illustrate the idea, a Second-Order Reduced Dispersion (SORD) marching scheme and Fourth-Order Reduced Dispersion (FORD) upwind scheme are developed. Computational tests demonstrate a favorable performance of the schemes. In spite of the previous practice restricting usage of even-order upwind schemes (fourth-order in particular), they exhibit a potential to stand among popular algorithms of computational hydraulics.

Natural Convection in a Cavity With a Wavy Wall Heated From Below and Uniformly Cooled From the Top and Both Sides

In this paper, natural convection inside a two-dimensional cavity with a wavy right vertical wall has been carried out. The bottom wall is heated by a spatially varying temperature and other three walls are kept at constant lower temperature. The integral forms of the governing equations are solved numerically using finite-volume method in the non-orthogonal body-fitted coordinate system. The semi-implicit method for pressure linked equation algorithm with higher-order upwinding scheme are used. The streamlines and isothermal lines are presented for three different undulations (1, 2 and 3) with different Rayleigh number and a fluid having Prandtl number 0.71. Results are presented in the form of local and average Nusselt number distribution for a selected range of Rayleigh number (100-106).

Natural Convection in a Rectangular Cavity Heated from Below and Uniformly Cooled from the Top and Both Sides

ABSTRACT In this article, a study of natural convection inside a rectangular cavity is carried out. The bottom wall is heated by a spatially varying temperature and other three walls are kept constant at lower temperature. The integral forms of the governing equations are solved numerically using the finite-volume method in a nonorthogonal body-fitted coordinate system. The SIMPLE algorithm with higher-order upwinding scheme is used. Results are presented in the form of local and average Nusselt number distribution for a range of Rayleigh numbers (100–106) and aspect ratios (0.5, 0.75, 1, 1.25, 1.5, and 2). The streamlines and isothermal lines are presented for various Rayleigh numbers and a fluid having Prandtl number 0.71.