A general relation for standing normal jumps in both hydraulic and dry granular flows

Steady free-surface flows can produce sudden changes in height and velocity, namely standing jumps, which demarcate supercritical from subcritical flows. Standing jumps have traditionally been observed and studied experimentally with water in order to mimic various hydraulic configurations, for instance in the vicinity of energy dissipators. More recently, some studies have emerged that investigate standing jumps formed in flows of dry granular materials, which are relevant to the design of protection dams against avalanches. In the present paper, we present a new explicit relation for the prediction of the height of standing jumps. We demonstrate the robustness of the new relation proposed by revisiting and cross-comparing a great number of data sets on standing jumps formed in water flows on horizontal and inclined smooth beds, in water flows on horizontal rough beds, and in flows of dry granular materials down smooth inclines. Our study reveals the limits of the traditional one-to-one relation between the sequent depth ratio of the jump and the Froude number of the incoming supercritical flow, namely the Bélanger equation. The latter is a Rankine–Hugoniot relation which does not take into account the gravitational and frictional forces acting within the jump volume, over the jump length, as well as the possible density change across the jump when the incoming fluid is compressible. The newly proposed relation, which is exact for grains and a reasonable approximation for water, can solve all of these issues. However, this relation can predict the height of the standing jump only if another length scale, namely the length of the jump, is known. We conclude our study by discussing empirical but simple closure relations to get a reasonable estimate of the jump length for water flows and dry granular flows. These closure relations can be used to feed the general jump relation and then predict with accuracy the heights of the jumps in a number of situations, provided that well-calibrated friction laws – described in the present study – are considered.