Direct numerical simulations of the shear flow over assemblies of uniformly sized, solid spheres attached to a flat wall have been performed using the lattice-Boltzmann method. The random sphere assemblies comprised monolayers, double layers and triple layers. The Reynolds number based on the sphere radius and the overall shear rate was much smaller than 1. The results were interpreted in terms of the drag force (the force in the streamwise direction) and lift force (the force in the wall-normal direction) experienced by the spheres as a function of the denseness of the bed and the depth of the spheres in the bed. The average drag and lift forces decay monotonically as a function of the surface coverage of the spheres in the top layer of the bed. The sphere-to-sphere variation of the drag and lift forces is significant due to interactions between spheres via the interstitial fluid flow.

The time-dependent behaviour of non-Boussinesq high-Reynolds-number density currents, released from a lock of height h0 and length x0 into a deep ambient and spreading over horizontal flat boundaries, is considered. We use two-dimensional Navier–Stokes simulations to cover: (i) a wide range of current-to-ambient density ratios, (ii) a range of length-to-height aspect ratios of the initial release within the lock (termed the lock aspect ratio λ = x0/h0) and (iii) the different phases of spreading, from the initial acceleration phase to the self-similar regimes. The Navier–Stokes results are compared with predictions of a one-layer shallow-water model. In particular, we derive novel insights on the influence of the lock aspect ratio (λ) on the shape and motion of the current. It is shown that for lock aspect ratios below a critical value (λcrit), the dynamics of the current is significantly influenced by λ. We conjecture that λcrit depends on two characteristic time scales, namely the time it takes for the receding perturbation created at the lock upon release to reflect back to the front, and the time of formation of the current head. A comparison of the two with space–time diagrams obtained from the Navier–Stokes simulations supports this conjecture. The non-Boussinesq effect is observed to be significant. While the critical lock aspect ratio (λcrit) is of order 1 for Boussinesq currents, its value decreases for heavy currents and increases significantly (up to about 20) for light currents. We present a simple analytical model which captures this trend, as well as the observation that for a light current the speed of propagation is proportional to λ1/4 when λ < λcrit.

During an attempt to work on a stratified flow problem envisaged as a sequel of the paper by Sameen & Govindarajan (2007), it was found that the original paper contained errors in §§ 3.4 and 4.3 due to a factor of iα, which was inadvertently missed in two places in the code (i) in the buoyancy term due to the use of vertical velocity and streamfunction interchangeably, and (ii) in the apportionment between kinetic and potential energy in the Gmax calculation. Because of this, there were significant differences in the effect of Grashof number on stability. Figure 1 is the modified figure 9 of the original paper, for Pr =7 and ΔT = 25 K. The Poiseuille–Rayleigh–Bénard mode appears at Gr = 39.12 and is seen not to merge with the Poiseuille mode, unlike the conclusion made earlier. This modification applies at any Prandtl number from 10−2 to 102. The corrected versions of figures 17 and 21, showing Gmax contours for different Pr at Gr = 0 and different Gr for Pr = 1, are plotted in figures 2 and 3, respectively. The large growth reported at β = 0 was thus erroneous. The other main conclusions of the paper, that Prandtl number changes transient growth qualitatively, but not the least stable eigenmode, whereas viscosity stratification, which has a huge impact on exponential growth/decay, does not change transient growth much, remain the same. The secondary instabilities also remain unchanged. The stability equations (3.2) to (3.4) in the paper should read (for explanation, please refer to Sameen & Govindarajan 2007) (1)(2)(3)