Flows in conduits or channels are of interest in science, engineering, and everyday life. Flows in closed conduits or channels, like pipes or air ducts, are entirely in contact with rigid boundaries. Most closed conduits in engineering applications are either circular or rectangular in cross section. Open-channel flows, on the other hand, are those whose boundaries are not entirely a solid and rigid material; the other part of the boundary of such flows may be another fluid, or nothing at all. Important open-channel flows are rivers, tidal currents, irrigation canals, or sheets of water running across the ground surface after a rain.
In both closed conduits and open channels, the shape and area of the cross section of the flow can change along the stream; such flows are said to be nonuniform. Flows are those that do not change in geometry or flow characteristics from cross section to cross section are said to be uniform. Remember that flows can be either steady (not changing with time) or unsteady (changing with time). In this chapter we will look at laminar and turbulent flows in conduits and channels. The emphasis in this chapter is on steady uniform flow in straight channels. That’s a simplification of flows in the natural world, in rivers and in the ocean, but it will reveal many fundamental aspects of those more complicated flows. The material in this chapter is applicable to a much broader class of flows, in pipes and conduits, as well; such matters are covered in standard textbooks on fluid dynamics.
This chapter focuses on two of the most important aspects of channel flow: boundary resistance to flow, and the velocity structure of the flow. The discussion is built around two reference cases: steady uniform flow in a circular pipe, and steady uniform flow down an inclined plane. Flow in a circular pipe is clearly of great practical and engineering importance, and it is given lots of space in fluid-dynamics textbooks. Flow down a plane is more relevant to natural Earth-surface settings (sheet floods come to mind), and it serves as a good reference for river flows.
The first section looks at laminar flow in a planar open channel, to derive expressions for the distributions of shear stress and velocity across the cross section. There are two equivalent ways of doing that: specializing the Navier–Stokes equations (which, remember, are a general statement of Newton’s second law as applied to fluid flows) to the given kind of flow, or writing Newton’s second law directly for the given kind of flow. We will take the second approach here. Then in further sections we will tackle the much more difficult problem of resistance and velocity in turbulent flows in pipes and channels. That will necessitate a deeper examination of the nature of shear stresses in turbulent flow, and a careful consideration of the differences between what I will call smooth flow and rough flow. The outcome will be some widely useful techniques as well as greatly increased understanding. The section on velocity distributions is intricate and lengthy, and may not seem as directly useful, but it reveals some really fundamental concepts.