Maximum kinetic energy dissipation and the stability of turbulent Poiseuille flow

Following Malkus’s (J. Fluid Mech., vol. 1, 1956, pp. 521–539) proposal that turbulent Poiseuille channel flow maximises total viscous dissipation $D$, a variety of variational procedures have been explored involving the maximisation of different flow quantities under different constraints. However, the physical justification for these variational procedures has remained unclear. Here we address more recent claims that mean flow viscous dissipation $D_{m}$ should be maximised on the basis of a statistical stability argument, and that maximising $D_{m}$ yields realistic mean velocity profiles (Malkus, J. Fluid Mech., vol. 489, 2003, pp. 185–198). We clarify the connection between maximising $D_{m}$ and other flow quantities, verify Malkus & Smith’s, (J. Fluid Mech., vol. 208, 1989, pp. 479–507) claim that maximising the ‘efficiency’ yields realistic profiles and show that, in contrast, maximising $D_{m}$ does not yield realistic mean velocity profiles as recently claimed. This leads us to revisit Malkus’s statistical stability argument for maximising $D_{m}$ and to address some of its limitations. We propose an alternative statistical stability argument leading to a principle of minimum kinetic energy for fixed pressure gradient, which suggests a principle of maximum $D$ for fixed Reynolds number under certain conditions. We discuss possible ways to reconcile these conflicting results, focusing on the choice of constraints.