# 9.1: Introduction

• • Contributed by Bill Smyth
• Professor (Department of Aerospace and Ocean Engineering) at Oregon State University

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:

1. The fluid is inviscid: $$ν$$ = 0.
2. The fluid is homogeneous: $$\rho = \rho_0$$.
3. The flow is confined to the $$x$$−$$z$$ plane, so there is no dependence on $$y$$ and no motion in the $$y$$ direction.
4. The channel width $$W$$ is uniform and the walls are vertical.
5. The flow has the character of gravity waves in the long-wave limit:
1. The horizontal (streamwise) velocity is independent of depth: $$\vec{u}=u(x,t)\hat{e}^{(x)}$$.
2. The pressure is hydrostatic: $$p = \rho_0g(\eta −z)$$.

Some of these assumptions will be relaxed as we go along. Figure $$\PageIndex{1}$$: (a) River flow over an obstruction. Photo by Jonathan Ball. (b) “The Bishop Wave”, a storm on the downwind slope of the Sierra Nevada mountain range. Wind is from right to left; clouds and dust at left indicate rapidly rising, turbulent air. Note the similarity to (a). (Photo by Robert Symons, 5 March 1950.) Further details are on page iii.