Development of gravity currents on slopes under different interfacial instability conditions

We present experimental results on the development of gravity currents moving onto sloping boundaries with slope angles theta = 7 degrees, 10 degrees and 15 degrees. Different regimes of flow development are observed depending on the slope angle and on the initial velocity and density profiles, characterized by the Richardson number J(i) = delta(i)g(0)'/Delta u(i)(2), where delta(i), Delta u(i) and g(0)' are, respectively, the velocity interface thickness, the maximum velocity difference and reduced gravity at the beginning of the slope. For J(i) > 0.7 and the larger slope angle, the flow strongly accelerates, reaches a maximum at the beginning of the Kelvin-Helmholtz instability, then decelerates and re-accelerates again. For 0.3 < J(i) < 0.6, instability occurs earlier and velocity oscillations are less. When J(i) <= 0.3 the increase in velocity is smooth. The magnitude of velocity oscillation depends on the combined effect of J(i) and slope angle, expressed by an overall acceleration parameter (T) over bar (a) = (delta(i)/U-i) ((U-c-U-i)/x(c)), which, to first order, is given by J(i) sin theta, where U-c and x(c) are, respectively, the velocity and position at instability onset. The velocity increases smoothly up to an equilibrium state when (T) over bar (a) <= 0.06 and exhibits an irregular behaviour at larger values of (T) over bar (a). The critical Richardson number J(c) decreases with increasing J(i) (increasing delta(i)/h(i)) which is due to wall effects and delta/delta(rho) not equal 1. After the beginning of Kelvin-Helmholtz instability, entrainment rates are close to those of a mixing layer, decreasing to values of a gravity current after the mixing layer reaches the boundary. It is shown here that the interfacial instability during current development affects the bottom shear stress which can reach values of c(D) approximate to 0.03 regardless of initial conditions. By solving numerically the depth integrated governing equations, the gravity flow velocity, depth and buoyant acceleration in the flow direction can be well predicted for all the performed experiments over the full measurement domain. The numerical results for the experiments with J(i) > 0.3 predict that the current requires a distance of at least x(n) approximate to 40h(i) to reach a normal state of constant velocity, which is much larger than the distance x(n) approximate to 10h(i) required in the case of a current with J(i) <= 0.3 that is commonly assumed for downslope currents.