4: Flow in Channels
- Page ID
- 4168
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
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.
- 4.1: Introduction to Flow in Channels
- 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.
- 4.2: Laminar Flow Down an Inclined Plane
- We apply Newton’s second law to steady and uniform flow down an inclined plane; the strategy is to look at a block of the flow, bounded by imaginary planes normal to the bottom, with unit cross-stream width and unit streamwise distance. Such a block of fluid is said to be a “free body”. Because the flow is assumed to be steady and uniform, all of the forces in the streamwise direction that are exerted upon the fluid within the free body at any given time must add up to be to zero.
- 4.3: Turbulent Flow in Channels - Initial Material
- If you made experiments with pipe flows and channel flows at very low Reynolds numbers, before the transition to turbulent flow, you would find beautiful agreement between theory and observation—something that is always satisfying for both the theoretician and the experimentalist. But for turbulent flows, which is the situation in most flows that are of practical interest, the story is different.
- 4.4: Turbulent Shear Stress
- If the material or property passively associated with the fluid is on the average unevenly distributed—if its average value varies in a direction normal to the mean motion—then the balanced turbulent transfer of fluid across the planes causes a diffusive transport or “flow” of this property, usually referred to as a flux, in the direction of decreasing average value. This kind of transport is called turbulent diffusion.
- 4.5: Structure of Turbulent Boundary Layers
- In the following section are some of the most important facts and observations on the turbulence structure of turbulent boundary layers—with steady uniform flow down a plane as a reference case, but the differences between this and other kinds of boundary-layer flow lie only in minor details and not in important effects.
- 4.6: Flow Resistance
- This section takes account of what is known about the mutual forces exerted between a turbulent flow and its solid boundary. You have already seen that flow of real fluid past a solid boundary exerts a force on that boundary, and the boundary must exert an equal and opposite force on the flowing fluid. It is thus immaterial whether you think in terms of resistance to flow or drag on the boundary.
- 4.7: Velocity Profiles
- You have already seen that the profile of time-average local fluid velocity from the bottom to the surface in turbulent flow down a plane is much blunter over most of the flow depth than the corresponding parabolic profile for laminar flow. This is the place to amplify and quantify the treatment of velocity profiles in turbulent boundary-layer flows.