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

Abstract

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.

Viscous flow in domains with corners: Numerical artifacts, their origin and removal

“…The asymptotics can be matched with the numerical solution outside a given radius in a variety of ways. This approach has proved successful in similar situations in which corner singularities exist [34]. However, it has been found that for our problem such an alteration of the scheme merely shifts the pressure cliff to the arc at the radius where the asymptotics has been applied.…”

An Asymptotic Fitting Finite Element Method with Exponential Mesh Refinement for Accurate Computation of Corner Eddies in Viscous Flows

“…Blum [6] discussed application of dual singular function method to semilinear biharmonic equations, such as the Navier-Stokes equations or the von Kármán equations, however, presenting the numerical results only for linear problems. Shi et al proposed a method that combines asymptotics of the solution and local mesh refinement near a corner for solution of the Navier-Stokes equations [34]. The authors of [34] mentioned using a local block mesh refinement, which seems to be equivalent to algebraic mesh refinement.…”

“…Shi et al proposed a method that combines asymptotics of the solution and local mesh refinement near a corner for solution of the Navier-Stokes equations [34]. The authors of [34] mentioned using a local block mesh refinement, which seems to be equivalent to algebraic mesh refinement.…”

“…It has been found out that local mesh refinement near corners and use of analytical formulas of asymptotic solution near corners produce better results for problems with corner singularities. Some of the popular techniques to overcome slow convergence are as follows: singular function method [19,35], singular complement method [2], dual singular function method [6,7,10,11], introduction of analytical constraints to finite element formulation [33], truncation of corners and introduction of Dirichlet-to-Neumann boundary conditions for domains with truncated corners [21], and other methods based on the similar ideas [26,34]. Also, various methods based solely on mesh refinement (without using asymptotic expansion of the solution) were developed (see, for example, [1,3,14,30,31]).…”

Solution of the planar Newtonian stick–slip problem with the singular function boundary integral method

“…These methods range from special mesh-reÿnement schemes to sophisticated techniques that incorporate, directly or indirectly, the form of the local asymptotic expansion, which is known in many occasions. An exhaustive survey of treatment of singularities in elliptic boundary value problems is provided in the recent articles by Li and Lu [1], Dosiyev [2] and Shi et al [3]. Knowledge of the coe cients appearing in the local solution expansion is often desired in many engineering applications.…”