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Numerical Methods By K Das Pdf
this study investigated the effect of inaccurate measurement of the large-scale features of a coarse mesh on the convergence rate of the finite element (fe) method applied to the flow of a complex turbulent mixture in a pipe. the high accuracy of the fe method can be ensured by adequately capturing the large-scale features of the fine mesh. a coarser mesh is a convenient tool for the size representation of an fe mesh, since it needs much less computational time than the fine mesh. however, the influence of the inaccuracy of the large-scale features of a coarse mesh on the convergence rate is not yet clear. this paper focused on the effect of the large-scale features of the coarse mesh on the convergence rate of the fe method. a mathematical model was developed based on the navier-stokes equations using a second-order finite difference method. a three-dimensional flow in a pipe under laminar flow conditions was simulated using the fe method. the simulation results from the fe method were compared with those of experimental results and those obtained with the statistical iterative multiphase flow technique (simpflow-2) using a fine mesh. based on the simulation results, the accuracy of the large-scale features of the coarse mesh was evaluated based on the statistics of the errors. these statistics were compared with those of the errors associated with the errors of the fine mesh. as predicted, the fe method exhibited a good convergence rate when the coarse mesh features matched those of the fine mesh. however, when the coarse mesh features differed from those of the fine mesh, the fe method exhibited a poor convergence rate. thus, it was concluded that the convergence rate of the fe method depends on the accuracy of the coarse mesh.
in this work, we present the first 3d eulerian sph code built at the national aeronautics and space administration (nasa) jpl. eulerian sph (esph) is a lagrangian finite-volume method, which does not require the reconstruction of the mesh in both space and time. therefore, esph has no restrictions on the grid density and size. the advantages of esph include: (1) the temperature has a good spatial resolution; (2) the code does not need to deal with advection problems for density distribution; (3) the maximum grid size is unlimited; (4) the stability of the numerical scheme is quite sensitive to some of the design factors such as the kernel. to simulate the flow in the neoprene rocket engine combustor using esph, the numerical method must satisfy the following requirements: (1) it should be sufficiently accurate to represent all the unsteady acoustic modes in the combustor; (2) it should provide a robust stability boundary to avoid a collapse of mesh; (3) it should maintain a reasonable computational efficiency by reducing the computational cost of advection terms and pressure smoothing process. by comparing the numerical results for different mesh sizes and distribution schemes, it is shown that the computation can be conducted on a typical size triangular mesh without a specific treatment on the mesh deformation by adjusting the distribution function. furthermore, the stability limit at low reynolds numbers indicates that esph is reliable in both low and high reynolds numbers regime, and it is also confirmed by the numerical results on the correlation of the diameter growth rate to the mesh density. 5ec8ef588b
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