Bounds on the momentum transport by turbulent shear flow in rotating systems

Abstract

Bounds on the momentum transport by laminar or turbulent shear flows between two parallel plates in constant relative motion in a rotating system are derived. The axis of rotation is parallel to the plates. The dimensionless component of the rotation vector perpendicular to the relative motion of the plate is denoted by the Coriolis number τ. Through the consideration of separate energy balances for the poloidal and the toroidal components of the fluid velocity field a variational problem is formulated in which τ enters as a parameter. Bounds that are derived under the hypothesis that the extremalizing vector fields are independent of the streamwise coordinate suggest that no state of turbulent motion can exist for $-2\sqrt{1708} \equiv -\hbox{\it Re}_E \leq \hbox{\it Re} \leq 1708/\tau + \tau$ with $\tau \gtrsim \sqrt{1708}$.