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Article

B.P. LEONARD and SIMIN MOKHTARI

In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving…

Abstract

In 1982, Smith and Hutton published comparative results of several different convection‐diffusion schemes applied to a specially devised test problem involving near‐discontinuities and strong streamline curvature. First‐order methods showed significant artificial diffusion, whereas higher‐order methods gave less smearing but had a tendency to overshoot and oscillate. Perhaps because unphysical oscillations are more obvious than unphysical smearing, the intervening period has seen a rise in popularity of low‐order artificially diffusive schemes, especially in the numerical heat‐transfer industry. This paper presents an alternative strategy of using non‐artificially diffusive higher‐order methods, while maintaining strictly monotonic transitions through the use of simple flux‐limiter constraints. Limited third‐order upwinding is usually found to be the most cost‐effective basic convection scheme. Tighter resolution of discontinuities can be obtained at little additional cost by using automatic adaptive stencil expansion to higher order in local regions, as needed.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 2 no. 5
Type: Research Article
ISSN: 0961-5539

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Article

Huanxin Lai, Gailan Xing, Shantong Tu and Ling Zhao

The purpose of this paper is to present a pressure‐correction procedure for incompressible flows using unstructured meshes. A method of implementing high‐order spatial…

Abstract

Purpose

The purpose of this paper is to present a pressure‐correction procedure for incompressible flows using unstructured meshes. A method of implementing high‐order spatial schemes on unstructured grids was introduced.

Design/methodology/approach

The procedure used a collocated cell‐centered unstructured grid arrangement. In order to improve the accuracy of calculation, the widely used high‐order schemes for convection, developed for structured grids and in the form of either the normalized variable and space formulation (NVSF) or the total variation diminishing (TVD) flux limiters (FL), were introduced and implemented onto the unstructured grids. This implementation was carried out by constructing a local coordinate and introducing a virtual upstream node.

Findings

The procedure was validated by calculating the lid‐driven cavity flows which had benchmark numerical solutions. For comparison, these flows were also computed by a commercial package, the FLUENT. The results obtained by the present procedure agreed well with the benchmark solution although very coarse grids were used. For the FLUENT, however, worse agreements with the benchmark solutions were obtained although the grids used for computation were the same. These demonstrated the robustness of the presented numerical procedure.

Originality/value

With the present method, high‐order schemes in either NVSF or TVD FL forms for structured grids can be easily implemented onto unstructured grids. This provides more choices of high‐order schemes for calculating complex flows.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 21 no. 3
Type: Research Article
ISSN: 0961-5539

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Article

G. Papadakis and G. Bergeles

A finite difference scheme for convection term discretization, calledBSOU (stands for Bounded Second Order Upwind), is developed and itsperformance is assessed against…

Abstract

A finite difference scheme for convection term discretization, called BSOU (stands for Bounded Second Order Upwind), is developed and its performance is assessed against exact or benchmark solutions in linear and non‐linear cases. It employs a flux blending technique between first order upwind and second order upwind schemes only in those regions of the flow field where spurious oscillations are likely to occur. The blending factors are calculated with the aid of the convection boundedness criterion. In all cases the scheme performed very well, minimizing the numerical diffusion errors. The scheme is transportive, conservative, bounded, stable and accurate enough so as to be suitable for inclusion into a general purpose solution algorithm.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

A. Pascau, C. Pérez and D. Sánchez

A new discretization scheme named NOTABLE (New Option forthe Treatment of Advection in the Boundary Layer Equations) ispresented. Despite its name, this scheme is intended…

Abstract

A new discretization scheme named NOTABLE (New Option for the Treatment of Advection in the Boundary Layer Equations) is presented. Despite its name, this scheme is intended to be used in a general transport equation to discretize the convective term. It is formally third‐order accurate in regions of smooth solution and first‐order accurate at grid points having local maxima. Within the finite‐volume formulation it relates the face values to the nodal values via a non‐linear function. This scheme has been compared with well‐known high‐order schemes like QUICK and it has always given more accurate solutions. After assessing the scheme in several unidimensional test cases for which an exact solution is available, its performance in a complex swirling flow is addressed.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

Mohamed Rady, Eric Arquis, Dominique Gobin and Benoît Goyeau

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated…

Abstract

Purpose

This paper aims to tackle the problem of thermo‐solutal convection and macrosegregation during ingot solidification of metal alloys. Complex flow structures associated with the development of channels segregate and sharp gradients in the solutal field call for the implementation of accurate methods for numerical modeling of alloy solidification. In particular, the solute transport equation is convection dominated and requires special non‐oscillarity type high‐order schemes to handle the regions of channels segregates.

Design/methodology/approach

In the present study, a time‐splitting approach has been adopted to separately handle solute advection and diffusion. This splitting technique allows the application of accurate total variation dimensioning (TVD) schemes for solution of solute advection. Applications of second‐order Lax‐Wendroff TVD SUPERBEE and fifth‐order weighted essentially non‐oscillatory (WENO) schemes are described in the present article. Classical numerical solution of solute transport using hybrid and central‐difference schemes are also employed for the purpose of comparisons. Numerical simulations for solidification of Pb‐18%Sn in a two‐dimensional rectangular cavity have been carried out using different numerical schemes.

Findings

Numerical results show the difficulty of obtaining grid‐independent solutions with respect to local details in the region of channels. Grid convergence patterns and numerical uncertainty are found to be dependent on the applied scheme. In general, the first‐order hybrid scheme is diffusive and under predicts the formation of channels. The second‐order central‐difference scheme brings about oscillations with possible non‐physical extremes of solute composition in the region of channel segregates due to sharp gradients in the solutal field. The results obtained using TVD and WENO schemes contain no oscillations and show an excellent capture of channels formation and resolution of the interface between solute‐rich and depleted bands. Different stages of channels formation are followed by analyzing thermo‐solutal convection and macrosegregation at different times during solidification.

Research limitations/implications

Accurate prediction of local variation in the solutal and flow fields in the channels regions requires grid refinement up to scales in the order of microscopic dendrite arm spacing. This imposes limitations in terms of large computational time and applicability of available macroscopic models based on classical volume‐averaging techniques.

Practical implications

The present study is very useful for numerical simulation of macrosegregation during ingot casting of metal alloys.

Originality/value

The paper provides the methodology and application of TVD schemes to predict channel segregates during columnar solidification of metal alloys. It also demonstrates the limitations of classical schemes for simulation of alloy solidification.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 20 no. 8
Type: Research Article
ISSN: 0961-5539

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Article

Hou Ping‐Li, Tao Wen‐Quan and Yu Mao‐Zheng

Based on the normalized variable diagram, the weakness of the Gaskell and Lau's convective boundedness criterion (GL‐CBC) is revealed by numerical example. By careful…

Abstract

Based on the normalized variable diagram, the weakness of the Gaskell and Lau's convective boundedness criterion (GL‐CBC) is revealed by numerical example. By careful consideration of the smoothness of the normalized variable variation pattern, more rigorous constraints on the interface value interpolation are found. A new CBC is thus proposed, whose feasibility and correctness are demonstrated by the inspection of ten existing bounded schemes and a numerical example.

Details

Engineering Computations, vol. 20 no. 8
Type: Research Article
ISSN: 0264-4401

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Article

Yedidi N. Murty

The effect of throughflow and Coriolis force on convective instabilities in micropolar fluid layer heated from below for free‐free, isothermal and micro‐rotation free…

Abstract

The effect of throughflow and Coriolis force on convective instabilities in micropolar fluid layer heated from below for free‐free, isothermal and micro‐rotation free boundaries is investigated. Calculations are made using a lower order Galerkin approximation to solve the eigenvalue problem for stationary instability. It is observed that both stabilizing and destabilizing factors due to constant vertical throughflow can be enhanced by rotation.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 9 no. 6
Type: Research Article
ISSN: 0961-5539

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Article

T. Bo, H. Iacovides and B.E. Launder

This paper presents finite volume computations of turbulent flow througha square cross‐sectioned U‐bend of curvature strong enough(Rc/D =0.65) to cause separation. A zonal…

Abstract

This paper presents finite volume computations of turbulent flow through a square cross‐sectioned U‐bend of curvature strong enough (Rc/D =0.65) to cause separation. A zonal turbulence modelling approach is adopted, in which the high‐Re k‐ε model is used over most of the flow domain with the low‐Re, I‐equation model of k‐transport employed within the near‐wall regions. Computations with grids of different sizes and also with different discretization schemes, demonstrate that for this flow the solution of the k and ε equations is more sensitive to the scheme employed in their convective discretization than the solution of the mean flow equations. To avoid the use of extremely fine 3‐Dimensional grids, bounded high order schemes need to be used in the discretization of the turbulence transport equations. The predictions, while encouraging, displayed some deficiencies in the downstream region due to deficiencies in the turbulence model. Evidently, further refinements in the turbulence model are necessary. Initial computations of flow and heat transfer through a rotating U‐bend, indicate that at rotational numbers (Ro = ΩD/Wb) relevant to blade cooling passages, the Coriolis force can substantially modify the hydrodynamic and thermal behaviour.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 1
Type: Research Article
ISSN: 0961-5539

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Article

H. Lai and Y.Y. Yan

In this paper the effects of choosing dependent variables and cell face velocities on convergence of the SIMPLE algorithm are discussed. Using different velocity…

Abstract

In this paper the effects of choosing dependent variables and cell face velocities on convergence of the SIMPLE algorithm are discussed. Using different velocity components as either dependent variables or cell‐face velocities, both convergent rate and calculation accuracy of the algorithm are compared and studied. A novel method, named “cross‐correction”, is developed to improve the convergence of the algorithm of using non‐orthogonal grids. Cases with benchmark and analytical solutions are used for numerical experiments and validation. The results show that, although different velocity components are employed as either dependent variables or cell face velocities, there is no obvious difference in both the convergent rates and numerical solutions. Moreover, the “cross‐correction” method is validated by computations with several first‐order and high‐order convection schemes; and the generality of convergence improvement achieved by the method is shown in the paper.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 11 no. 6
Type: Research Article
ISSN: 0961-5539

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Article

Gh. Juncu and I. Iliuta

The paper presents the numerical performance of the preconditionedgeneralized conjugate gradient (PGCG) methods in solvingnon‐linear convection — diffusion equations…

Abstract

The paper presents the numerical performance of the preconditioned generalized conjugate gradient (PGCG) methods in solving non‐linear convection — diffusion equations. Three non‐linear systems which describe a non‐isothermal chemical reactor, the chemically driven convection in a porous medium and the incompressible steady flow past a sphere are the test problems. The standard second order accurate centred finite difference scheme is used to discretize the models equations. The discrete approximations are solved with a double iterative process using the Newton method as outer iteration and the PGCG algorithm as inner iteration. Three PGCG techniques, which emerge to be the best performing, are tested. Laplace‐type operators are employed for preconditioning. The results show that the convergence of the PGCG methods depends strongly on the convection—diffusion ratio. The most robust algorithm is GMRES. But even with GMRES non‐convergence occurs when the convection—diffusion ratio exceeds a limit value. This value seems to be influenced by the non‐linearity type.

Details

International Journal of Numerical Methods for Heat & Fluid Flow, vol. 5 no. 3
Type: Research Article
ISSN: 0961-5539

Keywords

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