Upper bounds on general dissipation functionals in turbulent shear flows: revisiting the ‘efficiency’ functional

Abstract

We show how the variational formulation introduced by Doering & Constantin to rigorously bound the long-time-averaged total dissipation rate [ ] in turbulent shear flows can be extended to treat other long-time-averaged functionals lim sup_T→∞(1/T)×∫₀^Tf(D, D_m, D_v)dt of the total dissipation D, dissipation in the mean field D_m and dissipation in the fluctuation field D_v. Attention is focused upon the suite of functionals f = D(D_v/D_m)ⁿ and f = D_m(D_v/D_m)ⁿ (n [ges ] 0) which include the ‘efficiency’ functional f = D(D_v/D_m) (Malkus & Smith 1989; Smith 1991) and the dissipation in the mean flow f = D_m (Malkus 1996) as special cases. Complementary lower estimates of the rigorous bounds are produced by generalizing Busse's multiple-boundary-layer trial function technique to the appropriate Howard–Busse variational problems built upon the usual assumption of statistical stationarity and constraints of total power balance, mean momentum balance, incompressibility and boundary conditions. The velocity field that optimizes the ‘efficiency’ functional is found not to capture the asymptotic structure of the observed mean flow in either plane Couette flow or plane Poiseuille flow. However, there is evidence to suppose that it is ‘close’ to a neighbouring functional that may.