SUMMARYIn many cases, boundary integral equations contain a domain integral. This can be evaluated by discretization of the domain into domain elements. Historically, this was seen as going against the spirit of boundary element methods, and several methods were developed to avoid this discretization, notably dual and multiple reciprocity methods and particular solution methods. These involved the representation of the interior function with a set of basis functions, generally of the radial type. In this study, meshless methods (dual reciprocity and particular solution) are compared to the direct domain integration methods. The domain integrals are evaluated using traditional methods and also with multipole acceleration. It is found that the direct integration always results in better accuracy, as well as smaller computation times. In addition, the multipole method further improves on the computation times, in particular where multiple evaluations of the integral are required, as when iterative solvers are used. The additional error produced by the multipole acceleration is negligible.
SUMMARYA parallel computer implementation of a vorticity formulation for the analysis of incompressible viscous uid ow problems is presented. The vorticity formulation involves a three-step process, two kinematic steps followed by a kinetic step. The ÿrst kinematic step determines vortex sheet strengths along the boundary of the domain from a Galerkin implementation of the generalized Helmholtz decomposition. The vortex sheet strengths are related to the vorticity ux boundary conditions. The second kinematic step determines the interior velocity ÿeld from the regular form of the generalized Helmholtz decomposition. The third kinetic step solves the vorticity equation using a Galerkin ÿnite element method with boundary conditions determined in the ÿrst step and velocities determined in the second step. The accuracy of the numerical algorithm is demonstrated through the driven-cavity problem and the 2-D cylinder in a free-stream problem, which represent both internal and external ows. Each of the three steps requires a unique parallelization e ort, which are evaluated in terms of parallel e ciency.
SUMMARYThe evaluation of a domain integral is the dominant bottleneck in the numerical solution of viscous ow problems by vorticity methods, which otherwise demonstrate distinct advantages over primitive variable methods. By applying a Barnes-Hut multipole acceleration technique, the operation count for the integration is reduced from O(N 2 ) to O(N log N ), while the memory requirements are reduced from O(N 2 ) to O(N ). The algorithmic parameters that are necessary to achieve such scaling are described. The parallelization of the algorithm is crucial if the method is to be applied to realistic problems. A parallelization procedure which achieves almost perfect scaling is shown. Finally, numerical experiments on a driven cavity benchmark problem are performed. The actual increase in performance and reduction in storage requirements match theoretical predictions well, and the scalability of the procedure is very good.
The recent analytical of multi-layer analyses proposed by Sajjadi, Hunt and Drullion (2014) (SHD14 therein) is solved numerically for atmospheric turbulent shear flows blowing over growing (or unsteady) Stokes (bimodal) water waves, of low to moderate steepness. For unsteady surface waves the amplitude a(t) ∝ e kcit , where kc i is the wave growth factor, k is the wavenumber, and c i is the complex part of the wave phase speed, and thus the waves begin to grow as more energy is transferred to them by the wind. This will then display the critical height to a point where the thickness of the inner layer k i become comparable to the critical height kz c , where the mean wind shear velocity U (z) equals the real part of the wave speed c r . It is demonstrated that as the wave steepens further the inner layer exceeds the critical layer and beneath the cat's-eye there is a strong reverse flow which will then affect the surface drag, but at the surface the flow adjusts itself to the orbital velocity of the wave. We show that in the limit as c r /U * is very small, namely slow moving waves (i.e. for waves traveling with a speed c r which is much less than the friction velocity U * ), the energy-transfer rate to the waves, β (being proportional to momentum flux from wind to waves), computed here using an eddy-viscosity model, agrees with the asymptotic steady state analysis of Belcher and Hunt (1993) and the earlier model of Townsend (1980). The nonseparated sheltering flow determines the drag and the energy-transfer and not the weak critical shear layer within the inner shear layer. Computations for the cases when the waves are traveling faster (i.e. when c r > U * ) and growing
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