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Inviscid instability of the hyperbolic-tangent velocity profile-spectral "tau" solution

Bogusławski A.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

2001

Vol. 5, No 2

155-164

The paper presents a spectral solution of the Rayleigh equation for the case of parallel, free shear layer with the hyperbolic-tangent mean velocity profile. The expansion of the eigenfunction into the Chebyshev polynomial series allowed transformation of the differential eigenvalue problem into the general algebraic one. The standard algebraic eigenvalue problem was obtained by the use of Gary & Helgasson transformation. The results were compared with the shooting method. Although the calculations were carried out in order to validate the method, some additional study of the velocity ratio and momentum thickness influence on the temporal eigenmode growth rate was also performed.

Stress intensity factors computations using the singularity substraction technique incorporated with the Tau Method

Pan C. K, Liu K. M.

Computer Assisted Mechanics and Engineering Sciences

1998

Vol. 5, nr 1

55-64

In this paper we discuss the use of the singularity subtraction technique incorporated with the Tau Method for the numerical solution of singular partial differential equations which are relevant to the linear elastic fracture mechanics. To treat the singularity, we apply the singularity subtraction technique to the singular boundary value problems. The problems arising in this application are not in the standard form required by the Tau software. By introducing the pseudo-differential equations l k=0, k=1(1)m, to detrmine the stress intensity and higher order factors lk results in the standard boundary value problems. We consider two model crack problems including Motz ' anti-plane crack problem and plane strain problem defined by the biharmonic equation. We obtain results of considerable accuracy which compare favorably with those published in the recent literature.