Flow regimes in inclined open-channel flows of granular materials

Abstract

Open-channel flows of fluids may be classified as supercritical or subcritical, depending on whether their average velocity ῡ is larger, equal to or smaller than the propagation rate of small disturbances √(gh) cos α (where g is the gravitational acceleration, h is the flow depth and [alpha] is the channel inclination). Typically, the flow type is classified by the magnitude of the Froude number, ῡ√(gh), relative to its value under critical conditions Fr_c = √(cosα). Supercritical and subcritical flow represent conjugate states of open-channel flow; that is, a given supercritical flow will transition via an hydraulic jump that will transition a subcritical flow back to its corresponding supercritical value. flow. (However, energy considerations prohibit a reverse hydraulic jump that will transition a subcritical flow back to its corresponding supercritical value.) Supercritical flows are unaffected by downstream conditions, as they move faster than the downstream information can propagate upstream. Subcritical flows are strongly affected by downstream conditions. If downstream conditions are relaxed, a subcritical flow may transition back to supercritical flow (although not its conjugate state) via an expansion wave progagating upstream. The existence of a subcritical flow requires that the flow must be critical somewhere downstream before any abrupt expansion of the channel (such as the drop-off at the channel's end). The critical state prevents an expansion wave from propagating upstream from the expansion and causing a transition to supercritical flow. (A more complete discussion of these phenomena may be found in most introductory fluid mechanics textbooks; see, for example, Ref. 1, pp. 363-377.)