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In this paper, we analyse the behaviour of the solution of the Navier—Stokes equations near the corner of the driven cavity where the moving band touches the wall. At this point, the solution is singular. Since the singularity does not depend on the Reynolds number, it is sufficient to study the problem in the case of infinite viscosity, which is governed by the Stokes equations. We present an analytical asymptotic solution near the corner. Furthermore, numerical results are given, which were gained by an efficient multigrid algorithm. We will see that, for decreasing meshsize, the numerical solution converges to the derived analytical solution near the corner.

We study the sparse grid combination technique as an efficient method for the solution of fluid dynamics problems. The combination technique needs only O(h^–1_n(log(h^–1_n))^d–1) grid points for d‐dimensional problems, instead of O(h^–d_n) grid points used by the full grid method. Here, h_n = 2^–n denotes the mesh width of the grids. Furthermore, provided that the solution is sufficiently smooth, the accuracy (with respect to the L₂‐ and the L_∞‐norm) of the sparse grid combination solution is O(h^α_n(log(h^–1_n))^d–1), which is only slightly worse than O(h^α_n) obtained by the full grid solution. Here, α includes the order of the underlying discretization scheme, as well as the influence of singularities. Thus, the combination technique is very economic on both storage requirements and computing time, but achieves almost the same accuracy as the usual full grid solution. Another advantage of the combination technique is that only simple data structures are necessary. Where other sparse grid methods need hierarchical data structures and thus specially designed solvers, the combination method handles merely d‐dimensional arrays. Thus, the implementation of the combination technique can be based on any “black box solver”. However, for reasons of efficiency, an appropriate multigrid solver should be used. Often, fluid dynamics problems have to be solved on rather complex domains. A common approach is to divide the domain into blocks, in order to facilitate the handling of the problem. We show that the combination technique works on such blockstructured grids as well. When dealing with complicated domains, it is often desirable to grade a grid around a singularity. Graded grids are also supported by the combination technique. Finally, we present the first results of numerical experiments for the application of the combination method to CFD problems. There, we consider two‐dimensional laminar flow problems with moderate Reynolds numbers, and discuss the advantages of the combination method.

The purpose of this study is to examine theoretically the axisymmetric flow of a steady free-surface jet emerging from a tube for high inertia flow and moderate surface tension effect.

The method of matched asymptotic expansion is used to explore the rich dynamics near the exit where a stress singularity occurs. A boundary layer approach is also proposed to capture the flow further downstream where the free surface layer has grown significantly.

The jet is found to always contract near the tube exit. In contrast to existing numerical studies, the author explores the strength of upstream influence and the flow in the wall layer, resulting from jet contraction. This influence becomes particularly evident from the nonlinear pressure dependence on the upstream distance, as well as the pressure undershoot and overshoot at the exit for weak and strong gravity levels, respectively. The approach is validated against existing experimental and numerical data for the jet profile and centerline velocity where good agreement is obtained. Far from the exit, the author shows how the solution in the diffusive region can be matched to the inviscid far solution, providing the desired appropriate initial condition for the inviscid far flow solution. The location, at which the velocity becomes uniform across the jet, depends strongly on the gravity level and exhibits a non-monotonic behavior with respect to gravity and applied pressure gradient. The author finds that under weak gravity, surface tension has little influence on the final jet radius. The work is a crucial supplement to the existing numerical literature.

Given the presence of the stress singularity at the exit, the work constitutes a superior alternative to a computational approach where the singularity is typically and inaccurately smoothed over. In contrast, in the present study, the singularity is entirely circumvented. Moreover, the flow details are better elucidated, and the various scales involved in different regions are better identified.