Bucket covers for the buckets of a rotating component of a turbomachine are in radial opposition to a stationary component containing a circumferentially extending brush seal having bristles extending into engagement with sealing surfaces of the covers as the covers rotate by the bristles. The leading edges of the covers for the buckets in the plane of rotation and in the plane containing the brush seal have ramps or rounded surfaces which minimize the wear on the bristles and degradation of the performance of the seal as the bucket covers are rotated past the brush seal.