Unsteady Supersonic Flow for a Plate Undergoing Three-Stage Sinking Motion

A plate sinking in supersonic flow rapidly with pure acceleration was shown recently to have a flow structure and force behavior similar to those caused by a large-amplitude step motion, and this force evolution can be predicted by a nonlinear weighted Duhamel integral integrating a nonlinear solution due to large-amplitude forward step motions. In this paper, this previous study is extended to a more complex case, in which the plate sinks in supersonic flow with acceleration in the first stage and deceleration in the second stage, and stops sinking in the last stage. The flow structures are clarified numerically for each stage, according to whether the sinking is slow or fast compared to the horizontal wave speed, and the force-evolution behavior is displayed and related to the flow structures. It was found that a residual force exists during the motionless stage for fast sinking, due to the historical effect of wave evolution. In addition, although the nonlinear weighted Duhamel method predicts the force well for the first and second stages, it performs less well than a linear force method for the third stage, showing that a force method based on the forward step-motion model should be revised in the future.