Erick de Moraes Franklin

Subaqueous barchan dunes in turbulent shear flow. Part 1. Dune motion

Experiments are reported on the formation and migration of isolated dunes in a turbulent channel flow. These dunes have a very robust crescentic shape with horns pointing downstream, very similar to that of the barchan dunes observed in deserts at a much larger scale. Their main geometrical and dynamical properties are studied in detail, for four types of grains: the conditions for their formation, their morphology, the threshold shear stress for their motion, their velocity, erosion rate, minimum size and the longitudinal stripes of grains hollowed by fluid streaks in the boundary layer. In particular, the law for the dune velocity is found to involve two dimensionless parameters, the Shields number and the sedimentation Reynolds number, in contrast with predictions based on classical laws for particle transport. As the dune migrates, its size slowly decreases because of a small leakage of particles at the horn tips, and the erosion law is given. A minimum size is evidenced, which is shown to increase with the friction velocity and scale with a settling length.

Initial instabilities of a granular bed sheared by a turbulent liquid flow: length-scale determination

The transport of granular matter by a fluid flow is frequently found in nature and in industry. When the shear stresses exerted by the fluid flow on a granular bed are bounded to some limits, a mobile granular layer known as bed-load takes place, in which the grainsstay in contact with the fixed part of the granular bed. Under these conditions, an initiallyflat granular bed may be unstable, generating ripples and dunes, such as those observed indeserts, but also in pipelines conveying sand. There are evidences that these forms have atypical length correlated to their initial wavelength. So, the length-scale of the initial linear instabilities is a key point to understand the typical structures observed. This paper presents a theoretical study of the initial instabilities on a granular bed sheared by aturbulent liquid flow without free-surface effects, when bed-load is present. This studyconsists of a linear stability analysis, taking into consideration fluid flow, relaxation and gravity effects, and it is compared to published experimental data. It is proposed here,differently from many previous studies, that the initial wavelength of bed-forms varies with flow conditions when the fluid is a liquid.

Linear and nonlinear instabilities of a granular bed: Determination of the scales of ripples and dunes in rivers

Abstract Granular media are frequently found in nature and in industry and their transport by a fluid flow is of great importance to human activities. One case of particular interest is the transport of sand in open-channel and river flows. In many instances, the shear stresses exerted by the fluid flow are bounded to certain limits and some grains are entrained as bed-load: a mobile layer which stays in contact with the fixed part of the granular bed. Under these conditions, an initially flat granular bed may be unstable, generating ripples and dunes such as those observed on the bed of rivers. In free-surface water flows, dunes are bedforms that scale with the flow depth, while ripples do not scale with it. This article presents a model for the formation of ripples and dunes based on the proposition that ripples are primary linear instabilities and that dunes are secondary instabilities formed from the competition between the coalescence of ripples and free surface effects. Although simple, the model is able to explain the growth of ripples, their saturation (not explained in previous models) and the evolution from ripples to dunes, presenting a complete picture for the formation of dunes.

Morphology and displacement of dunes in a closed-conduit flow

Abstract The transport of solid particles entrained by a fluid flow is frequently found in industrial applications. A better knowledge of it, is of importance to improve particle related industrial processes. When shear stresses exerted by the fluid on the bed of particles are bounded to some limits, a mobile layer of particles known as bed-load takes place in which the particles stay in contact with the fixed bed. If it takes place over a non-erodible ground, and if the particle flow rate is small enough, an initial thin continuous layer of particles becomes discontinuous and composed of isolated dunes. We present here an experimental study to understand some features of the dynamics of isolated dunes under a fluid flow using a closed-conduit experimental loop made of transparent material. Acquired data concerns mainly dune morphology and displacement velocity under different conditions: different types of beads (diameters and densities) and different water flow conditions. We observed that an initial pile of beads placed in the conduit is rapidly deformed by the water flow, adopting a “croissant” shape, like barchan dunes found in deserts at a much larger scale. We observed also self-similarity in dunes dimensions and that dune displacement velocity scales with the inverse of their dimensions. The variation of the dune displacement velocity with the fluid shear velocity is discussed here.

Nonlinear instabilities on a granular bed sheared by a turbulent liquid flow

The granular media is of great importance in our quotidian, and their transport by a fluid flow is frequently found in nature and in industry. When the shear stresses exerted by the fluid flow on a granular bed are bounded to some limits, a mobile granular layer known as bed-load takes place in which the grains stay in contact with the fixed part of the granular bed. Under these conditions, a flat granular bed may be unstable, generating ripples and dunes. In a recent article (Franklin, 2010), the mechanisms of this instability were explained and a linear stability analysis was presented, in which a scaling between the fluid flow conditions and the typical length of the initial bed-forms was proposed. The present paper proposes a nonlinear stability analysis (weakly nonlinear approach) applicable to sheared granular beds, shedding light on the evolution of the bed-forms after their initial phase. The scope of the nonlinear analysis is the same as that of Franklin (2010): granular beds under turbulent liquid flows and in the presence of bed-load. It is shown here that, in this case, the initial instabilities saturate (supercritical bifurcation). Also, a discussion is made on some published experimental data.

The formation and migration of sand ripples in closed conduits: Experiments with turbulent water flows

Abstract The transport of solid particles by a fluid flow is frequently found in nature and industry. Some examples are the transport of sand in rivers and hydrocarbon pipelines. When the shear stresses exerted by a fluid flow on a granular bed remain moderate, some grains are set in motion without fluidizing the bed; the moving grains form a layer, known as bed load, that moves while maintaining contact with the fixed part of the bed. Under bed load conditions, the granular bed may become unstable, generating ripples and dunes. Sand ripples are commonly observed in closed conduits and pipes such as in petroleum pipelines, sewer systems, and dredging lines. Although of importance for many scientific domains and industrial applications, the formation of ripples in closed conduits is not well understood, and the problem is still open. This paper presents an experimental study on the formation and migration of sand ripples under a turbulent closed-conduit flow and bed-load conditions. In our experiments, fully-developed turbulent water flows were imposed over a granular bed of known granulometry in a transparent channel, and bed load took place. For different water flow rates and grain diameters, the growth and migration of bedforms were filmed by a high-definition camera, and a numerical code was developed to determine the wavelength and celerity of the bedforms from the acquired images. The obtained results are compared with published stability analyses.

The feedback effect caused by bed load on a turbulent liquid flow

Experiments on the effects due solely to a mobile granular layer on a liquid flow are presented (feedback effect). Nonintrusive measurements were performed in a closed conduit channel of rectangular cross section where grains were transported as bed load by a turbulent water flow. The water velocity profiles were measured over fixed and mobile granular beds of same granulometry by Particle Image Velocimetry. The spatial resolution of the measurements allowed the experimental quantification of the feedback effect. The present findings are of importance for predicting the bed-load transport rate and the pressure drop in activities related to the conveyance of grains.Experiments on the effects due solely to a mobile granular layer on a liquid flow are presented (feedback effect). Nonintrusive measurements were performed in a closed conduit channel of rectangular cross section where grains were transported as bed load by a turbulent water flow. The water velocity profiles were measured over fixed and mobile granular beds of same granulometry by Particle Image Velocimetry. The spatial resolution of the measurements allowed the experimental quantification of the feedback effect. The present findings are of importance for predicting the bed-load transport rate and the pressure drop in activities related to the conveyance of grains.

Length scale of density waves in the gravitational flow of fine grains in pipes

Gravitational flow of grains in pipes is frequently encountered in industry. When the grains and pipes are size-constrained, granular flow may result in density waves consisting of alternate high- and low-compactness regions. This paper discusses the length scale of density waves that appear when fine grains fall vertically in pipes. A one-dimensional model and a linear stability analysis of the model are presented. The analysis suggests the presence of long-wavelength instability for the most unstable mode, moreover, a cutoff wavenumber from which the length scale is estimated. Finally, the model results are compared to experimental data.

The perturbation of a turbulent boundary layer by a two-dimensional hill

Turbulent boundary layers over flat walls in the presence of a hill are frequently found in nature and industry. Some examples are the airflows over hills and desert dunes, and also water flows over aquatic dunes inside closed conduits. The perturbation of a two-dimensional boundary layer by a hill introduces new scales in the problem, changing the way in which velocities and stresses are distributed along the flow. When in the presence of sediment transport, the stress distribution along the hill is strongly related to bed instabilities. This paper presents an experimental study on the perturbation of a fully developed turbulent boundary layer by a two-dimensional hill. Water flows were imposed over a hill fixed on the bottom wall of a closed conduit and the flow field was measured by particle image velocimetry. From the flow measurements, mean and fluctuation fields were computed. The general behaviors of velocities and stresses are compared to published asymptotic analyses and the surface shear stress is analyzed in terms of instabilities of a granular bed.

Birth of a subaqueous barchan dune

Barchan dunes are crescentic shape dunes with horns pointing downstream. The present paper reports the formation of subaqueous barchan dunes from initially conical heaps in a rectangular channel. Because the most unique feature of a barchan dune is its horns, we associate the time scale for the appearance of horns to the formation of a barchan dune. A granular heap initially conical was placed on the bottom wall of a closed conduit and it was entrained by a water flow in turbulent regime. After a certain time, horns appear and grow, until an equilibrium length is reached. Our results show the existence of the time scales 0.5t_{c} and 2.5t_{c} for the appearance and equilibrium of horns, respectively, where t_{c} is a characteristic time that scales with the grains diameter, gravity acceleration, densities of the fluid and grains, and shear and threshold velocities.