Flow Depth Profiles Research Articles

Determining the Flow Resistance of Racks and the Resulting Flow Dynamics in the Channel by Using the Saint-Venant Equations

Racks retain debris in wastewater treatment plants and shield sensitive machinery in numerous engineering applications. In open-channel flows, racks impound the upstream water level by posing local obstacles in the flow. Based on experimental investigations, empirical approaches usually predict the flow resistance by relying on Bernoulli’s energy principle. Since this principle does not correctly consider downstream conditions such as submerged flows, we present a more accurate workflow to determine the flow resistance based on the Saint-Venant equations. We will demonstrate how the loss coefficient and the hydraulic head loss are determined more reliably without adding complexity to the engineering realisation. In contrast to relying solely on two cross-sections with Bernoulli’s energy principle, applying the Saint-Venant equations enables determining the flow depth profile and the flow velocity in the entire channel. This workflow additionally allows predicting the channel’s hydraulic capacity and freeboard in arbitrary applications.

Applicability of Kinematic and Diffusive models for mud-flows: a steady state analysis

The paper investigates the applicability of Kinematic and Diffusive Wave models for mud-flows with a power-law shear-thinning rheology. In analogy with a well-known approach for turbulent clear-water flows, the study compares the steady flow depth profiles predicted by approximated models with those of the Full Dynamic Wave one. For all the models and assuming an infinitely wide channel, the analytical solution of the flow depth profiles, in terms of hypergeometric functions, is derived. The accuracy of the approximated models is assessed by computing the average, along the channel length, of the errors, for several values of the Froude and kinematic wave numbers. Assuming the threshold value of the error equal to 5%, the applicability conditions of the two approximations have been individuated for several values of the power-law exponent, showing a crucial role of the rheology. The comparison with the clear-water results indicates that applicability criteria for clear-water flows do not apply to shear-thinning fluids, potentially leading to an incorrect use of approximated models if the rheology is not properly accounted for.

The dam-break problem for eroding viscoplastic fluids

Natural gravity-driven flows can increase in volume by eroding the bed on which they descend. This process is called basal entrainment and is thought to play a key role in the bulk dynamics of geophysical flows. Although its study is difficult using field measurements, basal entrainment is more easily amenable to analysis using laboratory experiments. We studied basal entrainment by conducting dam-break experiments releasing a fixed amount of viscoplastic fluid (a Herschel–Bulkley fluid) on a sloping, erodible bed of fixed depth. Entrainment was observed continuously, far from the sidewalls, using cameras. Bed material was quickly entrained, which led to flow advancement. Although the slope inclination had clear effects on the entrainment mechanisms, as shown by the internal measurements, this did not translate into faster front progression. Instead, the depth and length of the entrainable material were the most important controlling parameters of front velocity, as the surge scoured out the entrainable layer, pushing the entrainable material downstream and following the rigid bed’s geometry. Bulk measurements (front position and flow depth profile) were also compared with predictions from lubrication theory.

Basal entrainment by Newtonian gravity-driven flows

Gravity-driven flows can erode the bed along which they descend and increase their mass by a factor of 10 or more. This process is called “basal entrainment.” Although documented by field observations and laboratory experiments, it remains poorly understood. This paper examines what happens when a viscous gravity-driven flow generated by releasing a fixed volume of incompressible Newtonian fluid encounters a stationary layer (composed of fluid with the same density and viscosity). Models based on depth-averaged mass and momentum balance equations deal with bed-flow interfaces as shock waves. In contrast, we use an approach involving the long-wave approximation of the Navier-Stokes equations (lubrication theory), and in this context, bed-flow interfaces are acceleration waves that move quickly across thin stationary layers. The incoming flow digs down into the bed, pushing up downstream material, thus advancing the flow front. Extending the method used by Huppert [“The propagation of two-dimensional and axisymmetric viscous gravity currents over a rigid horizontal surface,” J. Fluid Mech. 121, 43–58 (1982)] for modeling viscous dam-break waves, we end up with a nonlinear diffusion equation for the flow depth, which is solved numerically. Theory is compared with experimental results. Excellent agreement is found in the limit of low Reynolds numbers (i.e., for flow Reynolds numbers lower than 20) for the front position over time and flow depth profile.

On the stability of gradually varying mud-flows in open channels

In the present paper the stability of gradually varying mud-flows in wide open channel is analyzed. The governing equations derive from a Saint–Venant-like approach with a Herschel & Bulkley model for the fluid rheology. In order to define the stability of a given initial accelerating or decelerating flow depth profile, the spatial evolution of a wavefront is analyzed. A stability criterion based on the linearized flow model is introduced, showing that the streamwise non-uniformity of the flow substantially influences the stability limits. A positive flow depth slope induces a stabilization while a negative one determines a destabilizing effect. The dependency of the linear stability on the values of rheological parameters is deeply discussed. A non-linear analysis of the wavefront propagation, along with the full non-linear solution of the problem, has been carried out to confirm the role played by the initial profile on the flow stability. The results of the presented study may be useful for engineering applications in order to prevent or to inhibit roll-waves formation in subcritical flows of a Herschel & Bulkley fluid.

Flume investigation of landslide debris–resisting baffles

Landslide debris is a common occurrence in mountainous regions around the world that can potentially result in disastrous consequences to downstream facilities. Flow-impeding structures are often constructed along the flow path to impede this hazardous phenomenon. Baffles are a type of flow-impeding structure regularly installed using empirical and prescriptive design methods as the interaction mechanism and the influence of baffle configuration on flow impedance is not well understood. A series of flume experiments were carried out to investigate flows characterizing landslide debris impacting an array of baffles using dry uniform sand. The influence of baffle height, row number, and spacing between successive rows was examined. Photoconductive sensors were used to estimate flow velocity, laser sensors were installed to measure flow depth profiles, and high-speed cameras were used to capture flow kinematics. Experimental results reveal that baffles can be categorized relative to the approach flow depth (h) and increasing the baffle height from 0.75h to 1.5h leads to a 40% increase in upstream flow depths from backwater effects, more effective development of subcritical conditions, and additional energy losses of up to 9%. Increasing the number of rows of 1.5h baffles from a single row to a three-row staggered array results in up to 72% additional energy loss. The energy loss is attributed to the deflection of granular jets and additional backwater effects. Increasing the row spacing from 50 to 100 mm results in up to a 14% increase in energy loss.

The dam-break problem for concentrated suspensions of neutrally buoyant particles

AbstractThis paper addresses the dam-break problem for particle suspensions, that is, the flow of a finite volume of suspension released suddenly down an inclined flume. We were concerned with concentrated suspensions made up of neutrally buoyant non-colloidal particles within a Newtonian fluid. Experiments were conducted over wide ranges of slope, concentration and mass. The major contributions of our experimental study are the simultaneous measurement of local flow properties far from the sidewalls (velocity profile and, with lower accuracy, particle concentration) and macroscopic features (front position, flow depth profile). To that end, the refractive index of the fluid was adapted to closely match that of the particles, enabling data acquisition up to particle volume fractions of 60 %. Particle migration resulted in the blunting of the velocity profile, in contrast to the parabolic profile observed in homogeneous Newtonian fluids. The experimental results were compared with predictions from lubrication theory and particle migration theory. For solids fractions as large as 45 %, the flow behaviour did not differ much from that of a homogeneous Newtonian fluid. More specifically, we observed that the velocity profiles were closely approximated by a parabolic form and there was little evidence of particle migration throughout the depth. For particle concentrations in the 52–56 % range, the flow depth and front position were fairly well predicted by lubrication theory, but taking a closer look at the velocity profiles revealed that particle migration had noticeable effects on the shape of the velocity profile (blunting), but had little impact on its strength, which explained why lubrication theory performed well. Particle migration theories (such as the shear-induced diffusion model) successfully captured the slow evolution of the velocity profiles. For particle concentrations in excess of 56 %, the macroscopic flow features were grossly predicted by lubrication theory (to within 20 % for the flow depth, 50 % for the front position). The flows seemed to reach a steady state, i.e. the shape of the velocity profile showed little time dependence.

The dam-break problem for viscous fluids in the high-capillary-number limit

Experiments were undertaken to investigate dam-break flows where a finite volume of highly viscous fluid (glucose with viscosity μ ≈ 350 Pa s) maintained behind a lock gate was released into a horizontal or inclined flume. The resulting sequence of flow-depth profiles was tracked using a three-dimensional visualization system. In the low-Reynolds-number and high-capillary-number limits, analytical solutions can be obtained from the Navier–Stokes equations using lubrication theory and matched asymptotic expansions. At shallow slopes, similarity solutions can also be worked out. While the variation in the front position scaled with time as predicted by theory for both horizontal and sloping flumes, there was a systematic delay in the front position observed. Moreover, taking a closer look at the experimental flow-depth profiles shows that they were similar, but they noticeably deviated from the theoretical similarity form for horizontal planes. For sloping beds, the flow-depth profile is correctly predicted provided that different scalings are used at shallow and large slopes.

The dam-break problem for Herschel–Bulkley viscoplastic fluids down steep flumes

In this paper we investigate the dam-break problem for viscoplastic (Herschel–Bulkley) fluids down a sloping flume: a fixed volume of fluid initially contained in a reservoir is released onto a slope and flows driven by gravitational forces until these forces are unable to overcome the fluid’s yield stress. Like in many earlier investigations, we use lubrication theory and matched asymptotic expansions to derive the evolution equation of the flow depth, but with a different scaling for the flow variables, which makes it possible to study the flow behavior on steep slopes. The evolution equation takes on the form of a nonlinear diffusion–convection equation. To leading order, this equation simplifies into a convection equation and reflects the balance between gravitational forces and viscous forces. After presenting analytical and numerical results, we compare theory with experimental data obtained with a long flume. We explore a fairly wide range of flume inclinations from 6° to 24°, while the initial Bingham number lies in the 0.07–0.26 range. Good agreement is found at the highest slopes, where both the front position and flow-depth profiles are properly described by theory. In contrast, at the lowest slopes, theoretical predictions substantially deviate from experimental data. Discrepancies may arise from the formation of unsheared zones or lateral levees that cause slight flow acceleration.

Emplacement of Martian rampart crater deposits

We present a basic continuum flow model for the emplacement of the distal rampart crater deposits on Mars. Assuming cylindrical symmetry, analytic solutions of the volume and momentum conservation equations yield time‐dependent flow thickness and velocity profiles. The momentum equation has an inertial term, and a frictional resistance term that is proportional to the local volumetric flow rate. We find that only a few basic physical processes are necessary to form sharp distal ramparts. First, there must be sufficient material to form a continuum overland flow. Then, for simple choices of boundary conditions at the source of the flow, distal ramparts form naturally due to the cylindrical geometry, the inertia of the flow, and local frictional resistance. THEMIS images of rampart deposits show that the ejecta flow velocities were relatively slow, being diverted by preexisting obstacles even within ∼0.2 crater radii from the rim of the parent crater. To develop inferences about rampart deposit emplacement, we require measurements of the crater radius and flow distance, and assumptions about the maximum source flow velocity and depth. We measured nine clearly expressed rampart deposits for craters with diameters of 3.4–17.0 km. The model provides estimates of the emplacement time, the rampart width, and the shape of the radial flow depth profiles. We find that modest velocities of ∼27–116 m s−1, maximum source flow depths of 10–40 m, and emplacement durations of 12–31 min adequately explain the morphologies of selected rampart craters.

Real-time measurement of in vitro flow by Fourier-domain color Doppler optical coherence tomography.

The possibility of measuring a full Doppler flow depth profile in parallel by use of frequency-domain optical coherence tomography is demonstrated. The method is based on a local phase analysis of the backscattered signal and allows for imaging of bidirectional Doppler flow. The Doppler frequency limit is 5 kHz for the presented measurements and is set by half of the frame rate of the CCD detector array. We measured the flow of 0.3-microm microspheres suspended in distilled water at controlled flow rates and in vitro human blood flow through a 200-microm capillary with a real-time color-encoded Doppler tomogram rate of 2-3/s.

Experimental study of steady and unsteady free surface flows with water-clay mixtures

The steady and unsteady free-surface flow of hyperconcentrated water-kaolinite clay mixtures is analysed experimentally. These mixtures exhibit non-Newtonian behaviour if volumetric concentrations of solid particles exceed about 10%. The viscosity-concentration correlations, determined by rotary viscometer, indicate that the Bingham model can be used to describe such fluids. Using the rheological parameters of this model, the water-clay laminar flow friction factors can be determined either by calculating the Reynolds and Hedstróm numbers, or by applying the concept of the extended Reynolds number. A quantitative analysis of the unsteady dam-break type of the free-surface flow is presented, giving a comparison of flow depth profiles and propagation rates between the investigated mixtures and water.