Hydraulic control of continuously stratified flow over an obstacle

Motivated by the laboratory experiments of Browand & Winant (Geophys. Fluid Dyn., vol. 4, 1972, pp. 29–53), a series of two-dimensional numerical simulations of flow past a cylinder of diameter $d$ are run for different values of the approach Froude number ${\mathit{Fr}}_{o} = U/ Nd$ between $0. 02$ and $0. 2$ at $\mathit{Re}= O(100)$. The observed flow is characterized by blocking and upstream influence in front of the cylinder and by relatively thin, fast jets over the top and bottom of the cylinder. This continuously stratified flow can be understood in terms of an inviscid non-diffusive integral inertia–buoyancy balance reminiscent of reduced-gravity single-layer hydraulics, but one where the reduced gravity is coupled to the thickness of the jets. The proposed theoretical framework describes the flow upstream of the obstacle and at its crest. The most important elements of the theory are the inclusion of upstream influence in the form of blocked flow within an energetically constrained depth range and the recognition that the flow well above and well below the active, accelerated layers is dynamically uncoupled. These constraints determine, through continuity, the transport in the accelerated layers. Combining these results with the observation that the flow is asymmetric around the cylinder, i.e. hydraulically controlled, allows us to determine the active layer thicknesses, the effective reduced gravity and thus all of the integral flow properties of the fast layers in good agreement with the numerically computed flows.