Solution of the unsteady Euler equations using an implicit dual-time method

An unfactored implicit tine-marching method for the solution of the unsteady two-dimensional Euler equations on deforming grids is described. The present work is placed into a multiblock framework and fits into the development of a generally applicable parallel multiblock how solver. The convective terms are discretized using an upwind total variation diminishing scheme, whereas the unsteady governing equations are discretized using an implicit dual-time approach. The large sparse linear system arising from the implicit time discretization at each pseudotime step is solved efficiently by using a conjugate-gradient-type method with a preconditioning based on. a block incomplete lower-upper factorization. Results are shown for a series of pitching airfoil test cases selected. from the AGARD aeroelastic configurations for the NACA 0012 airfoil. Comparisons with experimental data and previous published results are presented. The efficiency of the method is demonstrated by looking at the effect of a number of numerical parameters, such as the conjugate gradient tolerance and the size of the global time step and by carrying out a grid refinement study. Finally, a demonstration test case for the Williams airfoil(Williams, B. R., "An Exact Test Case for the Plane Potential Flow About Two Adjacent Lifting Aerofoils:" National Physical Lab., Aeronautical Research Council, Research Memorandum 3717, London, 1973) with an oscillating flap is presented, highlighting the capability of the grid deformation technique.