Stable Manifolds and the Transition to Turbulence in Pipe Flow

Lower-branch traveling waves and equilibria computed in pipe flow and other shear flows appear intermediate between turbulent and laminar motions. We take a step towards connecting these lower-branch solutions to transition by deriving a numerical method for finding certain special disturbances of the laminar flow in a short pipe. These special disturbances cause the disturbed velocity field to approach the lower-branch solution by evolving along its stable manifold. If the disturbance were slightly smaller, the flow would relaminarize, and if slightly larger, it would transition to a turbulent state.