NEW THEORETICAL CONSIDERATIONS ON THE CRITICAL FLOW IN A CIRCULAR CONDUIT (PART 2)

Abstract

This study constitutes the second part of a first study devoted to the critical flow in a smooth circular conduit. The first study published on this subject showed that two critical states occur in a circular conduit for a given diameter D and slope S₀. The first one is observed at shallow depths while the second one settles down at greater depths. The study considered the example of the smooth circular conduit of diameter D = 1m and concluded that when the slope S₀ is such that S₀ > 0.00183813, two critical states occur for two different discharges. Slopes that are less than this value do not generate any critical state of the flow. It was found that the slope S₀ = 0.00183813 corresponds to the smallest slope that causes a single critical state of the flow. The present study is interested in the influence of the diameter D on the variation of the critical depth as well as on the fate of the two critical states of the flow. The partially filled smooth circular conduit is still considered herein. This shows that there is a diameter D₁, the smallest, which generates only one critical state of the flow for a given slope of the conduit. All the conduit diameters D greater than D₁ generate two critical flow states, while diameters smaller than D₁ are the location of no critical flow. In addition, it has been demonstrated that the more the diameter of the conduit increases, the more the first critical state occurs at shallower depths. Conversely, the second critical state of the flow is observed at greater depths.

Using the appropriate theoretical relationship, it was possible to calculate the pair of values (D₁ ; S₀) which generates a single critical state of the flow. These values have been plotted graphically, which allowing to know whether the conduit is the seat of one or two critical states, or even of no critical state, from the known pair of values (D ; S₀).