Turbulent Wagner problem with transition

Wagner problem is originally concerned with inviscid flow and unsteady force due to a small step motion, or attaining of a small angle of attack, of an airfoil in an initially uniform flow and has been studied recently for inviscid flow with large amplitude step motion. Here we propose to consider turbulent Wagner problem for a plate that is initially covered with a mixed laminar-boundary layer on both sides and is set into step motion of small or large amplitude and in direction normal to the plate. The evolution of skin friction and transition region in time are examined numerically. It is found that transition region unexpectedly changes direction of movement for small amplitude of step motion while global transition or laminarization exists for large amplitude step motion. The significance of this study is twofold. First, the present study treated a new and interesting problem since it combines two problems of fundamental interests, one is Wagner problem and the other is boundary layer transition. Second, the present study appears to show that the pressure gradient normal to the airfoil and caused by discontinuous step motion may have subtle influence on transition and the mechanism of this influence deserves further studies.