Linear growth in two-fluid plane Poiseuille flow

In recent years a new paradigm has emerged in linear stability theory due to the recognition of the importance of non-normality in the Orr–Sommerfeld equation as derived from the method of normal modes. For single-fluid flows it has been shown that it is possible for the kinetic energy of certain stable mode combinations to grow transiently before decaying to zero. We look again at the linear stability of two-fluid plane Poiseuille flow in two dimensions, concentrating on transient growth and its dependence on the viscosity and depth ratio. The procedure is to solve the stability equations numerically and consider disturbances defined as a sum of the least stable eigenmodes (not just the least stable interfacial mode). It is found that the variational method used to find maximum growth cannot be based upon the kinetic energy of the flow only and that interface deflection must be included in the formulation. We show which modes are necessary for inclusion in the disturbance expression and find that the interfacial mode does not make a significant contribution to possible energy growth. We examine the magnitude of maximum growth and the nature of the disturbances that lead to this growth. The linear energy rate equation shows that at moderate Reynolds numbers the mechanism responsible for the largest two-fluid growth is transfer of energy from the basic flow via the Reynolds stresses. The energy transfer is facilitated by streamline tilting that can be seen at the channel walls or at the interface. A similar effect has been found in single-fluid plane Poiseuille flow.