Efficient numerical methods to compute unsteady subsonic flows on unstructured grids

Over the last four decades the increase in computer power and the advances in solver technology has resulted in an estimated reduction of 10 orders in magnitude to compute flow problems. However, to solve the instationairy Reynolds-averaged Navier-Stokes equations, even today, a massive amount of CPU time may be required. For example, to solve the unsteady flow around a wind turbine, a month of computing time may be required, even on a series of processors. This thesis, therefore, aims at reducing the computing times, especially for unsteady flow computations, by investigating two independent aspects: efficient distribution of grid nodes through grid adaptation and fast nonlinear solvers. With respect to an efficient distribution of grid points the goal is to develop an efficient refinement strategy for solution-based mesh adaptation that is straightforward to implement in existing CFD solvers with grid adapting capabilities. An efficient refinement strategy is required because most refinement strategies that can be found in the literature easily lead to over-refinement, are not fully automated or refine a fixed number of grid cells independent on accuracy considerations. Consequently, these refinement strategies are not guaranteed to yield computationally efficient flow computations. The approach followed is to develop a refinement strategy that aims at constructing a sharp upper bound for the error indicator distribution, for an a priori given number of grid cells. This is a useful approach because a sharp upper bound is a crude approximation of Dirac delta distribution, which is the most efficient distribution regarding computational costs and numerical accuracy. Furthermore, the possibility to a priori define the number of grid cells enables the user to optimize numerical accuracy for a given amount of CPU costs, useful when computing time is limited. The results presented in this thesis show a correlation between the sharpness of the error indicator distribution and numerical accuracy for two-dimensional subsonic flow computations. For three dimensional and supersonic flow computations there is less evidence for such a correlation. The results presented, however, show that the number of grid adaptations is in all cases a rather important parameter regarding the numerical accuracy, something that is not always mentioned in the literature. Finally, for all cases, with the number of grid adaptations carefully chosen, the adaptation strategy yields computationally efficient flow computations when one aims at engineering order of accuracies in the output quantities. With respect to fast nonlinear solvers the goal is to reduce computing times of unsteady flow problems by developing and implementing a Jacobian-free Newton Krylov (JFNK) solution method in a general purpose CFD solver. In addition to reducing the CPU costs compared to the original solver (classic nonlinear multigrid), the new solution method should be relatively insensitive to the various sources of stiffness typically encountered in flow problems, e.g. the Reynolds number, the Mach number and the maximum aspect ratio. Classic nonlinear multigrid typically has difficulties in tackling these stiffness sources such that computing times can dramatically and unexpectly increase for a different flow configuration. A Newton-based method comes natural for unsteady flow computations because the initial solution obtained from the previous time step is usually accurate enough to avoid nonlinear stall or divergence. Furthermore, a Krylov method, in combination with a suitable preconditioner, is less sensitive to the various sources of stiffness encountered in flow problems. An approximate factorization of the Jacobian, that approximately matches the target residual operator, is used to precondition the linear systems. This preconditioner is used because it enables rapid linear convergence and is easy to reuse for the sequences of linear systems that need to be solved. The results presented in this thesis show that a preconditioned JFNK algorithm enables a dramatic reduction in CPU costs compared to standard nonlinear multigrid: when the iteration error is reduced below the temporal error the reduction in CPU costs can be as large as a factor of 25.1. Furthermore, for the ranges tested, the performance of JFNK has been found to be rather insensitive to the stiffness induced by the Reynolds number, maximum aspect ratio, grid density, physical time step and Mach number. Also the complexity induced by fluid structure interaction is easily tackled by JFNK. In addition to the nonlinear solver the efficiency of the temporal discretization and deflation-preconditioning based on coarse grid vectors is addressed. In this it is shown that for engineering order of temporal accuracy, higher order implicit time integration reduces computing times compared to standard second order implicit time integration. Deflation-preconditioning based on coarse grid vectors can either enhance or reduce the overall performance for flow problems and is, therefore, considered liable. Deflation-preconditioning based on coarse grid vectors applied to a symmetric and nonsymmetric (convection)-diffusion equation always improves convergence.Aerospace Design, Integration & OperationsAerospace Engineerin