Mixed Boundary Layer Skin Friction and Heat Transfer With Abrupt Transition

In the form that is commonly published in introductory textbooks, the classical problem of skin friction and heat transfer for a mixed laminar–turbulent boundary-layer flow on a flat-plate with an abrupt transition is nonconservative in mass, momentum, and energy. By forcing continuity in momentum and enthalpy thicknesses, the textbook problem takes on the appearance of conserving momentum and energy. But, by doing so while retaining a turbulent virtual origin at the plate's leading edge, the textbook example omits necessary jumps in these quantities and violates conservation of mass, momentum, and energy in the flow. Here we modify this classical problem to satisfy conservation principles through the introduction of either concentrated mass/momentum/energy fluxes, at the top of the boundary layer and/or concentrated surface shear stress/heat fluxes at the bottom. Out of this simple analysis comes the intriguing idea of an entrainment flux or inflow at the top of the boundary layer over the transition region.