On the Numerical Treatment of Corner Singularity in the Vorticity Field

Floryan, J. M.; Czechowski, Leszek

doi:10.1006/jcph.1995.1094

Cited by 18 publications

(17 citation statements)

References 1 publication

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“…The results of Table VI show the good adequation of the approximation rate of the velocity with the one predicted by Equation (17). This illustrates the efficiency of the subtraction method and shows, in particular, that the overall numerical method gives an approximation rate for the velocity that is optimal since it matches the interpolation rate of the solution.…”

Section: Lid-driven Cavity Flowmentioning

confidence: 54%

Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

Botella

Peyret

2001

Numerical Methods in Fluids

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SUMMARYThe solution of fluid flow problems exhibits a singular behaviour when the conditions imposed on the boundary display some discontinuities or change in type. A treatment of these singularities has to be considered in order to preserve the accuracy of high-order methods, such as spectral methods. The present work concerns the computation of a singular solution of the Navier-Stokes equations using the Chebyshev-collocation method. A singularity subtraction technique is employed, which amounts to computing a smooth solution thanks to the subtraction of the leading part of the singular solution. The latter is determined from an asymptotic expansion of the solution near the singular points. In the case of non-homogeneous boundary conditions, where the leading terms of the expansion are completely determined by the local analysis, the high accuracy of the method is assessed on both steady and unsteady lid-driven cavity flows. An extension of this technique suitable for homogenous boundary conditions is developed for the injection of fluid into a channel. The ability of the method to compute high-Reynolds number flows is demonstrated on a piston-driven two-dimensional flow. Copyright

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Section: Lid-driven Cavity Flowmentioning

confidence: 54%

Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

Botella

Peyret

2001

Numerical Methods in Fluids

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“…As an alternative to the least-square matching, the exact matching strategy, which solves the same number of matching equations as the coe cients, namely n = m+1 in Equation (22), were applied in References [17,19]. Nevertheless, we found that this method is not robust since the convergence speed and the results can be very sensitive to the matching locations.…”

Section: Combined Analytical-numerical Methodsmentioning

confidence: 96%

“…Since the numerical scheme is not applied to region A, the local asymptotic solution prescribed there is equivalent to providing a boundary condition for the numerical solution of the remaining computational domain (B and C). Hence in the practical implementation, the local solution for the velocities (16) and (17) are applied only as boundary conditions at the interface between regions A and B, which can be called the Stokes inlet, rather than in the entire Stokes region (A). The coe cients and the prescribed analytical boundary conditions at the Stokes inlet are updated at each outer iteration.…”

Section: Combined Analytical-numerical Methodsmentioning

confidence: 99%

“…These methods except for those ad hoc treatments given in Reference [9] are useful in improving the global numerical accuracy in many cases where the numerical pollution arising from an angular singularity is locally bounded. Nevertheless, the error has a non-local character for the anti-symmetrical part of the solution of the Navier-Stokes equations [17], especially when strong convection is present [12]. Moreover, considering the limitation of the ÿnite arithmetic in the approximation of singularities, the numerical results will become progressively worse if the singular locations are approached.…”

Section: Introductionmentioning

confidence: 97%

“…This method was applied to treat the stress singularity in die swell problems in Stokes ows [6,7,18]. Several studies [17,19,20] have employed the local asymptotic solution for the Stokes ow to overcome the vorticity singularity in the numerical solution of the NavierStokes equations adopting the stream function-vorticity formulation. They assigned the local asymptotic solution for the Stokes ow in the neighbourhood of an angular point as the boundary condition for the ow in the remaining region, which was solved using a ÿnite-di erence scheme.…”

Section: Introductionmentioning

confidence: 99%

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A combined analytical–numerical method for treating corner singularities in viscous flow predictions

Shi

Breuer

Durst

2004

Numerical Methods in Fluids

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SUMMARYA combined analytical-numerical method based on a matching asymptotic algorithm is proposed for treating angular (sharp corner or wedge) singularities in the numerical solution of the Navier-Stokes equations. We adopt an asymptotic solution for the local ow around the angular points based on the Stokes ow approximation and a numerical solution for the global ow outside the singular regions using a ÿnite-volume method. The coe cients involved in the analytical solution are iteratively updated by matching both solutions in a small region where the Stokes ow approximation holds. Moreover, an error analysis is derived for this method, which serves as a guideline for the practical implementation. The present method is applied to treat the leading-edge singularity of a semi-inÿnite plate. The e ect of various in uencing factors related to the implementation are evaluated with the help of numerical experiments. The investigation showed that the accuracy of the numerical solution for the ow around the leading edge can be signiÿcantly improved with the present method. The results of the numerical experiments support the error analysis and show the desired properties of the new algorithm, i.e. accuracy, robustness and e ciency. Based on the numerical results for the leading-edge singularity, the validity of various classical approximate models for the ow, such as the Stokes approximation, the inviscid ow model and the boundary layer theory of varying orders are examined. Although the methodology proposed was evaluated for the leading-edge problem, it is generally applicable to all kinds of angular singularities and all kinds of ÿnite-discretization methods.

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Parallel adaptive mesh refinement for first‐order system least squares

Brezina

García

Manteuffel

et al. 2012

Numerical Linear Algebra App

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SUMMARY This paper develops new adaptive mesh refinement strategies for first‐order system least squares (FOSLS) in conjunction with algebraic multigrid (AMG) methods in the context of nested iteration (NI). The goal is to reach a certain error tolerance with the least amount of computational cost and nearly uniform distribution of the error over all elements. To accomplish this, the refinement decisions at each refinement level are determined on the basis of minimizing the ‘accuracy‐per‐computational‐cost’ efficiency (ACE) measure that takes into account both error reduction and computational cost. The NI‐FOSLS‐AMG‐ACE approach produces a sequence of refinement levels in which the error is equally distributed across elements on a relatively coarse grid. Once the solution is numerically resolved, refinement becomes nearly uniform. Accommodations of the ACE approach to massively distributed memory architectures involve a geometric binning strategy to reduce communication cost. Load balancing begins at very coarse levels. Elements and nodes are redistributed using parallel quadtree structures and a space‐filling curve, which automatically ameliorates load balancing issues at finer levels. Numerical results show that the NI‐FOSLS‐AMG‐pACE approach is able to provide highly accurate approximations to rapidly varying solutions at relatively low cost. Excellent weak and strong scalability are demonstrated on 4096 processors for problems with 15 million biquadratic elements.Copyright © 2012 John Wiley & Sons, Ltd.

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On the Numerical Treatment of Corner Singularity in the Vorticity Field

Cited by 18 publications

References 1 publication

Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

Computing singular solutions of the Navier–Stokes equations with the Chebyshev‐collocation method

A combined analytical–numerical method for treating corner singularities in viscous flow predictions

Parallel adaptive mesh refinement for first‐order system least squares

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