Hydraulic control of homogeneous shear flows

Abstract

If a shear flow of a homogeneous fluid preserves the shape of its velocity profile, a standard formula for the condition for hydraulic control suggests that this is achieved when the depth-averaged flow speed is less than (gh)1/2. On the other hand, shallow-water waves have a speed relative to the mean flow of more than (gh)1/2, suggesting that information could propagate upstream. This apparent paradox is resolved by showing that the internal stress required to maintain a constant velocity profile depends on flow derivatives along the channel, thus altering the wave speed without introducing damping. By contrast, an inviscid shear flow does not maintain the same profile shape, but it can be shown that long waves are stationary at a position of hydraulic control.

Hydraulic Control of a Highly Stratified Estuarine Front*

“…The steep drop-off in bathymetry and the presence of significant sand waves in the region, with typical heights of order 1 m (Kostaschuk and Luternauer 1989), may be sufficient to break up the local boundary layer effects. This assessment is also consistent with comparisons to Garrett and Gerdes (2003). Although caution must be taken in comparing the uniform-density flow described by Garrett and Gerdes (2003) with the stratified flow present in the Fraser River, such a comparison might suggest that significant shear would force the critical Froude number toward a value less than 1.…”

“…This assessment is also consistent with comparisons to Garrett and Gerdes (2003). Although caution must be taken in comparing the uniform-density flow described by Garrett and Gerdes (2003) with the stratified flow present in the Fraser River, such a comparison might suggest that significant shear would force the critical Froude number toward a value less than 1. The observations summarized in Fig.…”

“…The role played by viscous effects in modifying the inviscid two-layer hydraulic solutions, described by Armi and Farmer (1986), Farmer and Armi (1986), and others, is still an area of active research with much that is unclear. However, a comparison of the present observations with earlier studies, including loose analogy to the unstratified case described by Garrett and Gerdes (2003), suggests that viscous affects may play a reduced role at the front associated with the Fraser lift off. As such, the remainder of this paper proceeds with an analysis of the observational data within an inviscid framework.…”

“…Their results suggest that these viscous processes result in a downstream shift of the hydraulic control location. Similarly, Garrett and Gerdes (2003) found that shear within a singlelayer, uniform-density flow reduced the critical value of the Froude number to a value less than 1 and also shifted the control location in the downstream direction.…”

Critical conditions and composite Froude numbers for layered flow with transverse variations in velocity

“…Related behaviour was pointed out by Stern (1974) in connection with a singlelayer flow in a rotating channel with rectangular cross-section (for which the critical condition is w 0 (V 2 D) −1 (1 − V 2 /g D)dx = 0) and can also be deduced from the critical condition g d 0 V −2 dz = 1 for a homogeneous, free-surface flow with vertical shear (Garrett & Gerdes 2003). In both cases, a sluggish subinterval makes the entire flow subcritical.…”