Bounds on dissipation in three-dimensional planar shear flows: reduction to two-dimensional problems

Bounds on turbulent averages in 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 at each Reynolds number $ \textit{Re}$ by numerically optimising subject to a spectral constraint, which requires a quadratic integral to be non-negative for all possible velocity fields. Past authors have eased computations by enforcing the spectral constraint only for streamwise-invariant (2.5-D) velocity fields, assuming this gives the same result as enforcing it for three-dimensional (3-D) fields. Here, we compute optimal bounds over 2.5-D fields and then verify, without doing computations over 3-D fields, that the bounds indeed apply to 3-D flows. One way is to directly check that an optimiser computed using 2.5-D fields satisfies the spectral constraint for all 3-D fields. A second way uses a criterion we derive that is based on a theorem of Busse (1972 Arch. Ration. Mech. Anal., vol. 47, pp. 28–35) for energy stability analysis of models with certain symmetry. The advantage of checking this criterion, as opposed to directly checking the 3-D constraint, is lower computational cost and natural extrapolation of the criterion to large $ \textit{Re}$ . We compute optimal upper bounds on friction coefficients for the wall-bounded Kolmogorov flow known as Waleffe flow and for plane Couette flow. This requires lower bounds on dissipation in the first model and upper bounds in the second. For Waleffe flow, all bounds computed using 2.5-D fields satisfy our criterion, so they hold for 3-D flows. For Couette flow, where bounds have been previously computed using 2.5-D fields by Plasting & Kerswell (2003 J. Fluid Mech., vol. 477, pp. 363–379), our criterion holds only up to moderate $ \textit{Re}$ , so at larger $ \textit{Re}$ we directly verify the 3-D spectral constraint. Over the $ \textit{Re}$ range of our computations, this confirms the assumption by Plasting & Kerswell that their bounds hold for 3-D flows.

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