Ségolène Mejean, Thierry Faug, and Itai Einav
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.
Key words: granular media, hydraulics, surface gravity waves
J. Fluid Mech. (2017), vol. 816, pp. 331–351. c Cambridge University Press 2017 doi:10.1017/jfm.2017.82