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This paper presents results of a direct numerical simulation ͑DNS͒ of turbulent flow over the rough bed of an open channel. We consider a hexagonal arrangement of spheres on the channel bed. The depth of flow has been taken as four times the diameter of the spheres and the Reynolds number has been chosen so that the roughness Reynolds number is greater than 70, thus ensuring a fully rough flow. A parallel code based on finite difference, domain decomposition, and multigrid methods has been used for the DNS. Computed results are compared with available experimental data. We report the first-and second-order statistics, variation of lift/drag and exchange coefficients. Good agreement with experimental results is seen for the mean velocity, turbulence intensities, and Reynolds stress. Further, the DNS results provide accurate quantitative statistics for rough bed flow. Detailed analysis of the DNS data confirms the streaky nature of the flow near the effective bed and the existence of a hierarchy of vortices aligned with the streamwise direction, and supports the wall similarity hypothesis. The computed exchange coefficients indicate a large degree of mixing between the fluid trapped below the midplane of the roughness elements and that above it.

SUMMARYThis paper presents a study of the performance of the non-linear co-ordinate transformations in the numerical integration of weakly singular boundary integrals. A comparison of the smoothing property, numerical convergence and accuracy of the available non-linear polynomial transformations is presented for two-dimensional problems. E ectiveness of generalized transformations valid for any type and location of singularity has been investigated. It is found that weakly singular integrals are more e ciently handled with transformations valid for end-point singularities by partitioning the element at the singular point. Further, transformations which are excellent for CPV integrals are not as accurate for weakly singular integrals. Connection between the maximum permissible order of polynomial transformations and precision of computations has also been investigated; cubic transformation is seen to be the optimum choice for single precision, and quartic or quintic one, for double precision computations. A new approach which combines the method of singularity subtraction with non-linear transformation has been proposed. This composite approach is found to be more accurate, e cient and robust than the singularity subtraction method and the non-linear transformation methods.

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