Bounds on dissipation in 3-D shear flows: reduction to lower-dimensional flows

Bounds on mean dissipation or transport by turbulent shear flows can be derived from the Navier-Stokes equations by a mathematical approach called the background method. Bounds that are optimal within this method can be computed numerically by solving an optimization problem subject to a so-called spectral constraint, which requires a quadratic integral to be non-negative for all admissible velocity fields. For computational tractability, past authors have assumed that enforcing the spectral constraint only for streamwise-invariant velocity fields gives the same result as enforcing it for fully 3D fields. This talk presents two ways of checking this assumption without performing any 3D computations: by checking the 3D spectral constraint a posteriori, and by applying a theorem of Busse (1972) for the energy stability problem. The first approach is more broadly applicable, but the second gives results that extrapolate more naturally to large Re. This talk will show applications of both approaches, including optimal bounds on dissipation for the wall-bounded Kolmogorov flow known as Waleffe flow, and a confirmation that the the optimal bounds for planar Couette flow reported by Plasting and Kerswell (2003) are indeed valid for fully 3D flows.