An accurate gradient and Hessian reconstruction method for cell‐centered finite volume discretizations on general unstructured grids

Betchen, Lee J.; Straatman, Anthony G.

doi:10.1002/fld.2050

Cited by 21 publications

(22 citation statements)

References 8 publications

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“…(16) and (17) until convergence one would hope that a first-order accurate gradient ∇ d∞ would be obtained. As reported in [19], convergence of this procedure is not guaranteed and underrelaxation may be needed, leading to a large number of required iterations and great computational cost. In practice, the desired accuracy will be reached with a finite number of iterations, but this number is not known a priori and must be determined by trials; furthermore, in order to maintain the property that grid refinement improves the accuracy, the number of iterations should increase with grid refinement in order to avoid accuracy stagnation.…”

Section: Unstructured Gridsmentioning

confidence: 99%

“…In the literature, usually the gradient discretisation is only briefly discussed within an overall presentation of a FVM, with only a relatively limited number of studies devoted specifically to it (e.g. [6,[16][17][18][19][20]). This suggests that existing gradient schemes are deemed satisfactory, and in fact there seems to be a widespread misconception that the DT and LS schemes are second-order accurate on any type of grid.…”

Section: Introductionmentioning

confidence: 99%

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A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

et al. 2017

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Finite volume methods (FVMs) constitute a popular class of methods for the numerical simulation of fluid flows. Among the various components of these methods, the discretisation of the gradient operator has received less attention despite its fundamental importance with regards to the accuracy of the FVM. The most popular gradient schemes are the divergence theorem (DT) (or Green-Gauss) scheme, and the least-squares (LS) scheme. Both are widely believed to be secondorder accurate, but the present study shows that in fact the common variant of the DT gradient is second-order accurate only on structured meshes whereas it is zeroth-order accurate on general unstructured meshes, and the LS gradient is second-order and first-order accurate, respectively. This is explained through a theoretical analysis and is confirmed by numerical tests. The schemes are then used within a FVM to solve a simple diffusion equation on unstructured grids generated by several methods; the results reveal that the zeroth-order accuracy of the DT gradient is inherited by the FVM as a whole, and the discretisation error does not decrease with grid refinement. On the other hand, use of the LS gradient leads to second-order accurate results, as does the use of alternative, consistent, DT gradient schemes, including a new iterative scheme that makes the common DT gradient consistent at almost no extra cost. The numerical tests are performed using both an in-house code and the popular public domain PDE solver OpenFOAM. This is the accepted version of the article published in:

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Section: Unstructured Gridsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

et al. 2017

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“…It yields the first derivatives of the particle distribution functions to second‐order accuracy and the second derivatives to first‐order accuracy . Very recently, Betchen and Straatman presented a new method for an accurate gradient and Hessian reconstruction for cell‐centered FV discretization. These solution gradients are used to extrapolate the distribution function values to the virtual upwind nodes as defined in Reference up to second‐order accuracy in space.…”

Section: Introductionmentioning

confidence: 99%

Two‐dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

Patil

Lakshmisha

2011

Numerical Methods in Fluids

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SUMMARY In this article, an extension to the total variation diminishing finite volume formulation of the lattice Boltzmann equation method on unstructured meshes was presented. The quadratic least squares procedure is used for the estimation of first‐order and second‐order spatial gradients of the particle distribution functions. The distribution functions were extrapolated quadratically to the virtual upwind node. The time integration was performed using the fourth‐order Runge–Kutta procedure. A grid convergence study was performed in order to demonstrate the order of accuracy of the present scheme. The formulation was validated for the benchmark two‐dimensional, laminar, and unsteady flow past a single circular cylinder. These computations were then investigated for the low Mach number simulations. Further validation was performed for flow past two circular cylinders arranged in tandem and side‐by‐side. Results of these simulations were extensively compared with the previous numerical data. Copyright © 2011 John Wiley & Sons, Ltd.

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“…In the present work, the values of the limiter ‰ are computed according to the procedure of Venkatakrishnan [27]. The cell-centered gradients appearing in Equation (69), as well as the components of the cell-centered Hessian tensors, are computed in the present work on the basis of the Gauss' theorem approach of Betchen and Straatman [28], which provides a second-order accurate approximation to the gradient field and a first-order accurate approximation to the components of the Hessian tensor. As discussed in that work, this approach provides an estimate of the gradient field that is robust with respect to the quality of the grid employed.…”

Section: Methodsmentioning

confidence: 99%

“…The advected value φ ip of the dependent variable is approximated by the limited second‐order upwind estimate

\begin{matrix} leftalign \\ rightalign-odd φ_{ip} \approx & align-even \frac{1}{2} (1 + \frac{{m ̇}_{ip}}{|{m ̇}_{ip}|}) (φ_{P} + {Ψ_{P} \nabla φ|}_{P} \cdot (x_{ip} - x_{P})) & rightalign-label & align-label \\ rightalign-odd & align-even + \frac{1}{2} (1 - \frac{{m ̇}_{ip}}{|{m ̇}_{ip}|}) (φ_{nb, ip} + {Ψ_{nb, ip} \nabla φ|}_{nb, ip} \cdot (x_{ip} - x_{nb, ip})) . & rightalign-label (69) \end{matrix}

In the present work, the values of the limiter Ψ are computed according to the procedure of Venkatakrishnan . The cell‐centered gradients appearing in Equation , as well as the components of the cell‐centered Hessian tensors, are computed in the present work on the basis of the Gauss' theorem approach of Betchen and Straatman , which provides a second‐order accurate approximation to the gradient field and a first‐order accurate approximation to the components of the Hessian tensor. As discussed in that work, this approach provides an estimate of the gradient field that is r...…”

Section: Numerical Experimentsmentioning

confidence: 99%

A computational method for geometric optimization of enhanced heat transfer devices based upon entropy generation minimization

Betchen

Straatman

2012

Numerical Methods in Fluids

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SUMMARY A computational fluid dynamics‐based optimization methodology is developed, appropriate for the geometric optimization of enhanced heat transfer devices based upon the principle of entropy generation minimization, in which the objective function is evaluated from a flow field obtained by computational simulation. A quasi‐Newton optimization procedure is employed, with computation of the objective function gradients based upon a finite difference approach. The optimization procedure is developed to be general with regard to the choice of objective function, the details of the problem under consideration, and the computational methodology employed in solving the fluid flow and heat transfer problems. A novel implementation of a Taylor series‐based procedure for the fast solution of nearby problems is presented, which is found to greatly benefit the efficiency of the present methodology. Finally, a numerical experiment is presented, illustrating the use of the present method in the geometric optimization of a practical enhanced heat transfer device on the basis of the criterion of entropy generation minimization. The optimization of the fin spacing of a simple plate fin heat sink is considered, and a comparison of the computational results with results obtained by analytical optimization based upon empirical friction factor and Nusselt number correlations is given. Copyright © 2012 John Wiley & Sons, Ltd.

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An accurate gradient and Hessian reconstruction method for cell‐centered finite volume discretizations on general unstructured grids

Cited by 21 publications

References 8 publications

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

A critical analysis of some popular methods for the discretisation of the gradient operator in finite volume methods

Two‐dimensional flow past circular cylinders using finite volume lattice Boltzmann formulation

A computational method for geometric optimization of enhanced heat transfer devices based upon entropy generation minimization

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