Unification of variational principles for turbulent shear flows: The background method of Doering-Constantin and the mean-fluctuation formulation of Howard-Busse

The recent discovery of a new variational formulation for bounding turbulent flow quantities has rekindled interest in the whole field of producing rigorous inequality results directly from the Navier-Stokes equations initiated by Howard and Busse in the 1960s. The new Background method introduced by Doering and Constantin (1992, 1993, 1995, 1996) for bounding turbulent viscous dissipation appears to differ so fundamentally from that of Howard and Busse that any connection between the two methods has been fascinatingly unclear. Recently highly suggestive results obtained asymptotically (Kerswell, 1997) have indicated that the variational problems produced for both cases of shear flow and convection may be equivalent. Here we confirm this for the shear flow case by uncovering the common underlying functional which both approaches seek to make stationary. Busse's upper bound problem is found to be the optimal dual or complementary variational problem to that which emerges from the Doering-Constantin approach. This ensures that Busse's best bound exactly coincides with that available within the Doering-Constantin formalism for all Reynolds numbers. (C) 1998 Elsevier Science B.V.