9.1: Introduction
- Page ID
- 18085
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Our third example of the application of the Navier-Stokes equations to natural flows is hydrostatic flows over topography. These occur in many natural settings such as downslope windstorms (Figure \(\PageIndex{1}\)b), tidal ocean currents and flow over dams. They are perhaps most recognizable in the flow of a small stream over a rocky bottom (Figure \(\PageIndex{1}\)a). For this discussion, we abandon the assumption that the amplitude of the disturbance is small. Instead, we assume that the flow is in hydrostatic balance, similar to the long-wave limit discussed above.
Assumptions:
- The fluid is inviscid: \(ν\) = 0.
- The fluid is homogeneous: \(\rho = \rho_0\).
- The flow is confined to the \(x\)−\(z\) plane, so there is no dependence on \(y\) and no motion in the \(y\) direction.
- The channel width \(W\) is uniform and the walls are vertical.
- The flow has the character of gravity waves in the long-wave limit:
- The horizontal (streamwise) velocity is independent of depth: \(\vec{u}=u(x,t)\hat{e}^{(x)}\).
- The pressure is hydrostatic: \(p = \rho_0g(\eta −z)\).
Some of these assumptions will be relaxed as we go along.
