Turbulence in gravity-driven film flow is usually discussed in terms of three-dimensional solitary-wave pulses as they are frequently observed in flows along smooth walls. Here, we show that ...free-surface turbulence arises in films along rippled substrates, as they are commonly employed in process engineering applications, already at rather low Reynolds numbers from the irregular break-up of the solitary-wave fronts. Short waves in the capillary regime replace the broken solitary waves beyond a certain Reynolds number. The crossover coincides with the occurrence of steady three-dimensional surface patterns upstream, which suppress travelling waves. The waves show power spectral densities with power-law exponents typical for weak capillary-wave turbulence. With increasing Reynolds number, the steepness of the power law declines to lower levels.
This paper presents an extensive uniflow cyclone dedusting study and analysis of respective performance data based on separation efficiency, pressure drop and cut size. A laboratory-scale uniflow ...cyclone with axial swirl vanes is used to test a range of setups employed by the industry. The most important swirl vane and vortex finder parameters, as well as concentration levels are examined in detail with reference to geometric changes. Comprehensive separation data is provided, showing the impact of angle of attack, core size ratio, blade overlap, vortex finder length and vortex finder diameter on performance levels. Grade efficiency curves and cut sizes for each setup are analyzed. A cut size below 4 μm is achieved for marble dust with a particle density of 2700 kg m−3. Results show that the cut size can be reduced by over 50% with an increase of pressure drop by approximately 30% in relation to the base configuration. For a steep angle of attack of 15° and a core size ratio of 0.125, the cut size can be reduced by 25%, with a 35% reduction of pressure drop.
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•Extensive experimental dedusting study with uniflow cyclones.•Cut size reduced by more than 25% with a reduction of pressure drop of 35%.•The performance increases with steeper angles of attack and higher core size ratios.•The optimal vortex finder length is found between 2 and 3 times the cyclone radius.
We study the incipient motion of single spheres in steady shear flow on regular substrates at low particle Reynolds numbers. The substrate consists of a monolayer of regularly arranged fixed beads, ...in which the spacing between beads is varied resulting in different angles of repose and exposures of the particle to the main flow. The flow-induced forces and the level of flow penetration into the substrate are determined numerically. Since experiments in this range had shown that the critical Shields number is independent of inertia but strongly dependent on the substrate geometry, the particle Reynolds number was fixed to 0.01 in the numerical study. Numerics indicates that rolling motion is always preferred to sliding and that the flow penetration is linearly dependent on the spacing between the substrate particles. Besides, we propose an analytical model for the incipient motion. The model is an extension of Goldman’s classical result for a single sphere near a plain surface taking into account the angle of repose, flow orientation with respect to substrate topography and shielding of the sphere to the linear shear flow. The effective level of flow penetration is the only external parameter. The model, applied to triangular and quadratic arrangements with different spacings, is able to predict the dependence of the critical Shields number on the geometry and on the orientation of the substrate. The model is in very good agreement with numerical results. For well-exposed particles, we observed that the minimum critical Shields number for a certain angle of repose does not depend sensitively on the considered arrangement. At large angles of repose, as expected in fully armoured beds, the model is consistent with experimental results for erodible beds at saturated conditions.
We study experimentally the impact of substrate topology on shear-flow-induced motion of a single bead at low particle Reynolds numbers. The substrates are regular quadratic and triangular ...arrangements of fixed spherical particles. Their topology is varied by using different spacings between the spheres. Here, we show that it has a strong impact not only on the critical Shields number for incipient bead motion but also on its motion above threshold. We focus on Shields numbers where the bead velocity is smaller than the settling velocity. For the different substrates, the data on the average bead velocity collapse on a master curve, showing the impact of the critical Shields number on the bead motion. To describe the bead motion, we develop a model for creeping flows based on expressions by Goldman, Cox and Brenner for the flow-induced forces and torques on a moving bead near a plane. Our model considers rolling and sliding motion. The bead detaches from the substrate on the downhill side at larger substrate spacing or higher Shields numbers, and flies through the interstices of the substrate until hitting the neighbouring substrate spheres. While sliding has only a minor effect on the average bead velocity, detachment has a strong impact. At large substrate spacings, it leads to a bistability, usually associated with inertial flows, even for adhesionless particles under creeping-flow conditions. The model shows good agreement with the experimental results.
The effect of concentration and molar ratio of cetyltrimethylammonium bromide with sodium salicylate (CTAB/NaSal) micellar aqueous solutions on the settling of non-Brownian spheres is determined. The ...micelles cause shear thinning, which may result in a shear-stress plateau. The measurements were performed in the wide range of Weissenberg numbers showing drag reduction and enhancement. Maximum drag coincides with the minimum relaxation time. For equimolar solutions, a correlation between Weissenberg number and normalized drag correction factor is proposed. The average settling velocity decreases proportionally to the inverse CTAB/NaSal concentration squared and is closely related to the micellar network size. Salt excess in 10 mM CTAB solutions results in an exponential increase in the average velocity. We find steady and unsteady regimes for the particle motion. In equimolar solutions, the frequency decreases with the concentration, whereas the dominant frequency does not change with salt excess, but is associated with the shear-stress oscillations. We show that the unsteady particle motion is related to the rheological time-dependent properties of the micellar solutions.
We study nonlinear resonance in viscous gravity-driven films flowing over undulated substrates. Numerical solution of the full, steady Navier–Stokes equations is used to follow the emergence of the ...first few free-surface harmonics with increasing wall amplitude, and to study their parametric dependence on film thickness, inertia and capillarity. Bistable resonance is computed for steep enough bottom undulations. As an analytic approach, we apply the integral boundary-layer method and derive an asymptotic equation valid for rather thin films. The analysis recovers the key numerical findings and provides qualitative understanding. It shows that higher harmonics are generated by a nonlinear coupling of the wall with lower-order harmonics of the free surface. It also accounts for bistable resonance, and produces a minimum model whose solution is similar to that of the Duffing oscillator.
A linear stability analysis of a Newtonian liquid film flowing down an inclined wavy plane is carried out. It is studied how wavy bottom variations, which are long compared to the film thickness, ...modify the stability of the steady film flow with respect to that down a flat inclined plane. By allowing for rather moderate bottom variations, it shows the impact of geometric nonlinearities on the instability. In this case, the spatial growth of disturbances becomes dependent on the phase along the bottom wave. Averaging over the bottom variations, it is found that on a large scale the critical Reynolds number for the onset of surface waves is higher than that for a flat bottom. As in the case of a flat bottom, the instability occurs at long wavelength. Locally, however, at the steep slopes the critical Reynolds number is lower than for a flat incline. In a certain range of waves numbers and Reynolds numbers, shorter waves may be excited at the steep slopes and damped at the flat ones.
We study viscous gravity-driven films flowing over periodically undulated substrates. Linear analysis describes steady flow along small amplitude corrugations for films of arbitrary thickness. ...Solving the resulting system numerically, we demonstrate resonance (or, possibly, near resonance) and identify different behaviours for thin, intermediate and thick films. Approximating the leading-order velocity profile by the free surface value allows for an analytic solution, which – in the limit of high Reynolds numbers – recovers the different regimes and reveals the relevant physical mechanisms. Our results support the view that the resonance is associated with an interaction of the undulated film with capillary-gravity waves travelling against the mean flow direction. As a consequence, the resonance peak is attained under conditions that render the wave phase velocity equal to zero in the laboratory reference frame, and thus permit direct exchange of energy between the steadily deformed film and the free surface.
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