Sediment-laden Open-channel Flows Research Articles

Turbulent kinetic energy budget of sediment-laden open-channel flows: bedload-induced wall-roughness similarity

New experiments in highly turbulent, steady, subcritical and uniform water open-channel flows have been carried out to measure the mean turbulent kinetic energy (TKE) budget of sediment-laden boundary layer flows with two sizes (dp = 3 mm and 1 mm) of Plexiglas particles $({\rm relative\ density}\ = 1.192)$ . The experiments covered energetic sediment transport conditions (Shields number of $0.35 < \theta < 1.2$ ) ranging from non-capacity to full-capacity flows in bedload-to-suspension-dominated transport modes (suspension number of $0.5 < {w_s}/{u_\ast } < 1.3$ where ${w_s}$ is the settling velocity and $u_*$ is the friction velocity) and for weakly to highly inertial, finite size turbulence-particle conditions (Stokes number of 0.1 < St < 3.5 and dp/η > 10 where $\eta$ is the Kolmogorov length scale). It was shown that the effects of sediments on the TKE budget are very pronounced in all large particle experiments for which a bedload layer of several grain diameter thickness is developed above the channel bed. When compared with the corresponding reference clear-water flows, the TKE shear-production rate for the 3 mm particle flows is strongly reduced in the wall region corresponding to the bedload layer. This turbulence damping is seen to increase with sediment load until full capacity for flows with constant Shields value, as well as with Shields number value. Inside this damped TKE shear-production zone, a distinct peak of maximal turbulence production appears to coincide with the upper edge of the bedload layer delimited by a sharp gradient in mean sediment concentration. This vertically upshifted peak of TKE production is accompanied by an enhanced net downward oriented TKE flux when compared with the reference clear-water flows. The downward diffused TKE is found to act in the bedload layer as a local energy source in reasonable balance with the sediment transport term. The mechanism behind this downward TKE transport was further analysed on the basis of coherent flow structure dynamics controlled by ejection- and sweep-type events. The agreement between the height of downward directed mean TKE flux and the height below which sweep-type events dominate the Reynolds shear-stress contribution over ejections, revealed the leading role played by sweeps in mean TKE transport. This agreement holds for all reference clear-water flows supporting the well-known wall-roughness-induced dominance of the sweep contribution in turbulent, rough clear-water boundary layer flows. Furthermore, for all 3 mm particle flows, the two referred to transition levels were significantly and similarly upshifted to the upper edge of the bedload layer. Only for these sediment-laden flows, the bedload layer thickness is seen to exceed the wall-roughness sublayer of the reference clear-water flows. This supports a strong analogy between wall-roughness effects in clear-water flows and bedload layer effects in sediment-laden flows, on the mean TKE budget induced by a similarly modified coherent flow structure dynamics. The bedload layer-controlled wall roughness is finally confirmed by the good prediction of the wall-roughness parameter ks of the logarithmic velocity distribution. An empirical formulation fitting the presented measurements is presented, valid over the range of Shields number values covered herein.

Semi-analytical modeling of sediment-laden open-channel flows with the effects of stratification, hindered settling, and eddy viscosities.

This study proposes semi-analytical models for simultaneous distribution of fluid velocity and suspended sediment concentration in an open-channel turbulent flow using three kinds of eddy viscosities. Apart from the classical parabolic eddy viscosity which is based on a log-law velocity profile, we consider two recently proposed eddy viscosities based on the concept of velocity and length scales. To deal with the flows with high sediment concentration, several turbulent features such as the hindered settling mechanism and the stratification effect are incorporated in the model. The governing system of highly nonlinear differential equations is solved using the homotopy analysis method (HAM), which produces solutions in the form of convergent series. Numerical and theoretical convergence analyses are provided for all three types of eddy viscosities. The effects of parameters on the derived models are discussed physically. Experimental data on both dilute and non-dilute flows are considered to verify the HAM-based solutions. The effects of the stratification correction factor (β) and the turbulent Schmidt number (α) reveal that they should be determined optimally for applicability of the proposed models in terms of accurate prediction with data. This optimal procedure required further investigation of these parameters, and, thus, an analysis of β and α is carried out, which linked them with the particle diameter through particle settling velocity, reference fluid velocity, and reference sediment concentration by proposing regression equations. Furthermore, using the optimal values of the parameters, the proposed models corresponding to the eddy viscosities based on the exponentially decreasing turbulent kinetic energy function and von Karman's similarity hypothesis are seen to be superior to the model corresponding to a parabolic eddy viscosity. Finally, a comment on the HAM is made where it is observed that the method can remove the numerical singularity of the governing equations at the water surface, which arises because of the consideration of vanishing eddy viscosity thereat.

Predicting the vertical low suspended sediment concentration in vegetated flow using a random displacement model

Based on the Lagrangian approach, this study proposes a random displacement model (RDM) to predict the concentration of suspended sediment in vegetated steady open channel flow. Validation of the method was conducted by comparing the simulated results by using the RDM with available experimental measurements for uniform open-channel flows. The method is further validated with the classical Rouse formula. To simulate the important vertical dispersion caused by vegetation in the sediment-laden open channel flow, a new integrated sediment diffusion coefficient is introduced in this study, which is equal to a coefficient β multiplying the turbulent diffusion coefficient. As such, the RDM approach for sandy flow with vegetation was established for predicting the suspended sediment concentration in low-sediment-concentration flow with both the emergent and submerged vegetation. The study shows that the value of β for submerged vegetation flow is larger than that for emergent vegetation flow. The simulated result using the RDM is in good agreement with the available experimental data, indicating that the proposed sediment diffusion coefficient model can be accurately used to investigate the sediment concentration in vegetated steady open channel flow.

An Expression for Velocity Lag in Sediment-Laden Open-Channel Flows Based on Tsallis Entropy Together with the Principle of Maximum Entropy.

In the context of river dynamics, some experimental results have shown that particle velocity is different from fluid velocity along the stream-wise direction for uniform sediment-laden open-channel flows; this velocity difference has been termed velocity lag in the literature. In this study, an analytical expression for estimating the velocity lag in open-channel flows was derived based on the Tsallis entropy theory together with the principle of maximum entropy. The derived expression represents the velocity lag as a function of a non-dimensional entropy parameter depending on the average and maximum values of velocity lag from experimental measurements. The derived expression was tested against twenty-two experimental datasets collected from the literature with three deterministic models and the developed Shannon entropy-based model. The Tsallis entropy-based model agreed better with the experimental datasets than the deterministic models for eighteen out of the twenty-two total real cases, and the prediction accuracy for the eighteen experimental datasets was comparable to that of the developed Shannon entropy-based model (the Tsallis entropy-based expression agreed slightly better than the Shannon entropy-based model for twelve out of eighteen test cases, whereas for the other six test cases, the Shannon entropy-based model had a slightly higher prediction accuracy). Finally, the effects of the friction velocity of the flow, the particle diameter, and the particles’ specific gravity on the velocity lag were analyzed based on the Tsallis entropy-based model. This study shows the potential of the Tsallis entropy theory together with the principle of maximum entropy to predict the stream-wise velocity lag between a particle and the surrounding fluid in sediment-laden open-channel flows.

Vertical Sediment Concentration Distribution in High-Concentrated Flows: An Analytical Solution Using Homotopy Analysis Method

Transport of suspended sediment in open channel flow has an enormous impact on real life situations, viz. control and management of reservoir sedimentation, geomorphic evolution such as dunes, rivers, and coastlines etc. Transport entails advection and diffusion. Turbulent diffusion is governed by the concept of Fick’s law, which is based on the molecular diffusion theory, and the equation that represents the distribution of sediment concentration is the advection-diffusion equation. The study uses the existing governing equation which considers different phases for solid and fluid, and then couples the two phases. To deal with high-concentrated flow, sediment and turbulent diffusion coefficients are taken to be different from each other. The effect of hindered settling on sediment particles is incorporated in the governing equation, which makes the equation highly non-linear. This study derives an explicit closed-form analytical solution to the generalized one-dimensional diffusion equation representing the vertical sediment concentration distribution with an arbitrary turbulent diffusion coefficient profile. The solution is obtained by Homotopy Analysis Method, which does not rely on the small parameters present in the equation. Finally, the solution is validated by comparing it with the implicit solution and the numerical solution. A relevant set of laboratory data is selected to check the applicability of the model, and a close agreement shows the potential of the model in the context of application to high-concentrated sediment-laden open channel flow.

Stochastic formulation of particle kinetics in wall-bounded two-phase flows

This paper presents a generalized framework of stochastic modeling for particle kinetics in wall-bounded flow. We modified a reflected Brownian motion process and straightforwardly obtained a Kramers equation for particle probability density function (PDF). After the wall effects were accounted for as a drift from zero in the mean displacement and suppression in the diffusivity of a particle, an analytical solution was worked out for PDF. Three distinguishable mechanisms were identified to affect the profile of particle probability distribution: external forces, turbophoresis effect, and wall-drift effect. The proposed formulation covers the Huang et al. (2009) model of a wall that produces electrostatic repulsion force and van der Waals force, as well as Monte-Carlo solutions for the Peter and Barenbrug (2002) model under a variety of relaxation times. Moreover, it successfully reproduces the two patterns of particle concentration profiles observed in experiments of sediment-laden open-channel flows. The strength of the wall-drift effect was found to be connected with the interaction frequency between particle and wall. Further exploration of the relationship among flow turbulence, particle inertia, and particle concentration is worthwhile.

Influence of Lift Force on the Settling Velocities of Rotating Particles in Two-Dimensional Shear Flow

AbstractWe derive a simple theoretical model for estimating the change in the magnitude of the particle slip velocity and the deflection in the trajectory of a particle induced by the lift force acting on a freely rotating particle settling under gravity in a shear flow. The lift force on the particle in small, intermediate, and large particle Reynolds number, is studied. We have found that the magnitude of the slip velocity is always reduced by the lift force, implying reduced particle settling velocities. The reduction in the slip velocity and the angle of deflection are the greatest when the shear is the greatest and the particle Reynolds number is the smallest. However, these effects decrease as the particle Reynolds number becomes larger. The derived equations are applied to evaluate the effect of the lift force on particle transport within turbidity currents, on the entrainment of small particles embedded in the lower boundary of a shear flow, and on sediment-laden open-channel flow.

Discussion of “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by Erik A. Toorman

The author presented a challenging contribution to the topic of sediment-laden open channel flows. The theoretical considerations implied differences in time scales between turbulence tt and suspended sediment particles tp, although limited experimental evidence was available. Herein the discussers present some prototype measurements of turbulence and suspended sediment time scales conducted in a small estuary.

Discussion of “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by Erik A. Toorman

Discussion of “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by Erik A. Toorman

Closure to “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by Erik A. Toorman

Closure to “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by Erik A. Toorman

PIV and PTV measurements in hydro-sciences with focus on turbulent open-channel flows

PIV is one of the most popular measurement techniques in hydraulic engineering as well as in fluid sciences. It has been applied to study various turbulent phenomena in laboratory experiments related to natural rivers, e.g., bursting phenomena near the bed, mixing layers observed at confluences, wake turbulence around dikes and piers, and so on. In these studies, PIV plays important roles in revealing the space-time structure of velocity fluctuations and coherent vortices. This review article focuses particularly on the applications of PIV to turbulent open-channel flows, which have been conducted for the past decade in Hydraulics Laboratory of Kyoto University. In Section 2, we introduce our experimental setup and PIV/PTV algorithm. In Section 3, we apply the PIV measurements to reveal turbulence characteristics and coherent structures in open-channel flows as well as in vegetated canopy flows. For complex flow situations, various applications of PIV to compound open-channel flows and wind-induced water waves are considered to reveal coherent vortices. In Section 4, we discuss some advanced PIV measurements in open-channel flows. The free-surface-elevation fluctuations and velocity components were measured simultaneously with two sets of cameras to examine phase-averaged parameters of turbulence. A multi-layer scanning PIV was developed to reveal 3D turbulence structure in compound open-channel flows. Our discriminator PIV/PTV was applied successfully to sediment-laden open-channel flows and revealed the fluid/particle interaction and the relationship between coherent structures and sediment concentration. Finally, we conducted simultaneous measurements of velocity and dye concentration with a combination of PIV and LIF in vegetated open-channel flow, which enables us to examine turbulent scalar flux of a passive contaminant.

Sediment-induced buoyancy destruction and drag reduction in estuaries

This paper presents an analysis of drag reduction by buoyancy destruction in sediment-laden open channel flow. We start from the log-linear profile proposed by Barenblatt (Prikladnaja Matematika i Mekhanika, 17:261–274, 1953), extended with a second length scale to account for free surface effects. Upon analytical integration over the water depth, an expression for sediment-induced drag reduction is found in terms of an effective Chezy number, water depth, bulk Richardson number, and Rouse number. This relation contains one empirical/experimental coefficient, which was obtained from a large series of numerical experiments with a 1DV point model. Upon calibration of this model against field and laboratory observations, we tuned the turbulent Prandtl–Schmidt number and found an optimal value of σT = 2, consistent to observations by Cellino and Graf (ASCE, J Hydraulic Engineering, 125:456–462, 1999). All numerical results could be correlated with the simple relation \( C_{\text{eff}} = C_0 + 4\sqrt {g} hRi_{*} \beta \), which is valid for fine sediment suspensions under conditions typical in open channel flow.

Particle–turbulence interaction and local particle concentration in sediment-laden open-channel flows

The present study describes an experimental investigation on the effects of ejection and sweep motions to the local sediment concentration and turbulence modulation in suspended sediment-laden open-channel flows by using a combination between a discriminator particle tracking velocimetry (D-PTV) for sediment particles and a discriminator particle image velocimetry (D-PIV) for fluid tracers. Fluid tracers and sediment particles were discriminated by their occupied particle sizes, and the particle velocity and the fluid velocity were determined simultaneously. It is necessary to investigate coherent structure and particle–turbulence interaction in suspended sediment-laden open-channel flows for predicting the transport of suspended sediment in rivers. There are a lot of previous studies on velocity measurements and numerical simulations in these suspended sediment flows. However, there are little investigation on simultaneous measurements between particles and fluid in bursting phenomena of open-channel flow. So, this study focuses the particle–fluid interaction in the ejection and sweep events by using the discriminator PIV/PTV. The present study found that the turbulence in sediment-laden flows is enhanced or suppressed as compared with that of sediment-free (clear-water) flow. These enhancement and suppression of turbulence are determined by the critical particle diameter, which is correlated with the Kolmogoroff microscale rather than the macroscale of turbulence. The strength of the ejections and sweeps was changed by suspended sediment. The local sediment concentration became about 20–40% larger when the ejection motion occurred, whereas it became about 10–30% smaller when the sweep motion occurred.

Errata and Addendum for “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by E. A. Toorman

Errata and Addendum for “Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow” by E. A. Toorman

Vertical Mixing in the Fully Developed Turbulent Layer of Sediment-Laden Open-Channel Flow

Eulerian equations for the vertical flux and momentum of suspended particles in dilute sediment-laden open-channel flow in equilibrium have been derived using the two-fluid approach. Reynolds averaging has been applied in order to allow validation of individual terms with experimental data. Consideration of the various terms of the vertical momentum balance with experimental flume data indicates that the drag force has to be separated into a mean and a turbulent contribution with different timescales, respectively, the (gravitational) particle timescale and the integral turbulence timescale. The resulting formulation further provides a new theoretical closure for the turbulent Schmidt number.

VELOCITY AND CONCENTRATION DISTRIBUTIONS IN SEDIMENT-MIXED FLUID: AN APPROACH WITH MIXING LENGTH CONCEPT

ABSTRACT Mathematical models have been developed to predict the vertical velocity and sediment concentration distributions in a sediment-laden turbulent open channel flow taking into account a modified mixing length based on sediment-induced stratification. The coupled non-linear differential equations for vertical velocity and concentration arising out of the problem have been solved numerically and the results have been compared with the comprehensive exiting data. Comparison shows good agreement with the observed data, which proves the validity of the present model.

河床波上の浮遊砂を伴う流れにおける粒子・流体の相互作用と乱流変調に関する研究

This study describes experimental investigation on particle-fluid interaction and turbulence modulation in suspended sediment-laden open-channel flow over sand dunes by using a discriminator particle tracking velocimetory (D-PTV), which was developed by Nezu and Azuma (JHE, ASCE, 2004). In this D-PTV system, sediment particles and fluid tracers are discriminated accurately by their occupied particle size, and thus, the particle velocity Up and fluid velocity Uf can be measured simultaneously.As the results, significant interactions between sediment particles and water were observed in the separation bubble (a reverse flow occurs) and near the rettachment point behind dune. Furthermore, it was revealed that the turbulence intensity of fluid became smaller in sediment-laden flow than in clear water flow without sediment.

Analysis of Velocity Lag in Sediment-Laden Open Channel Flows

Laboratory experiments have recently confirmed that the streamwise particle velocity is largely less than that of the fluid in sediment-laden flows. This velocity lag is investigated analytically in the present study based on the drag force exerting on a particle in the presence of other neighbors. The normalized drag force or the hindrance coefficient is found generally dependent on the particle concentration, particle Reynolds number, and specific gravity. The velocity lag is then derived by relating the hindrance coefficient to the shear stress distribution for uniform sediment-laden open channel flows. The analysis shows that the profile of the velocity lag, when normalized by the shear velocity, is associated with the shear Reynolds number, dimensionless particle diameter, and specific gravity. For the dilute condition, the velocity lag distribution varies only with the shear Reynolds number, and the lag can be ignored if the shear Reynolds number is less than unity. The theoretical predictions are comparable to limited experimental results.

Two-Phase Flow Analysis of Concentration Profiles

Two-phase flow analysis is used to analyze sediment concentration profiles in uniform open-channel flows over flat, sediment-starved beds that have high concentrations of single-sized sediment. Two-phase flow analysis can explicitly incorporate the effects of particle-particle interactions and particle inertia. Conventional convection-diffusion modeling cannot directly represent these phenomena and are thus limited. Both the two-phase flow formulation and the convection-diffusion modeling are compared against experimental data collected in sediment-starved sediment-laden flows. The two-phase flow model is shown to simulate the effect of both particle-particle interactions and particle inertia in these experimental flows. Simple criteria are given to determine when particle-particle interactions and particle inertia are important in sediment-laden open-channel flows over a flat bed. The current two-phase approach requires empirical formulas of the turbulence quantities and further experimental and analytical work is necessary to develop improved models for the velocity distribution and turbulence quantities.

Dispersion in Sediment-Laden Stream Flow

A theory is developed to demonstrate the effects of sorptive exchange on the transport of a chemical in a sediment-laden open-channel flow. Based on the multiple-scale method of homogenization, a depth-averaged transport equation is deduced up to a long time-scale. The dispersion coefficient is the sum of a modified Taylor dispersion coefficient and a dispersion coefficient due to a finite rate of mass exchange between dissolved phase in the water column and sorbed phase on suspended particles. These coefficients are functions of the suspension number and the bulk solid-water distribution ratio. It is shown that, for sufficiently large particles and solid fractions, enhancement of the longitudinal dispersion coefficient due to the sorptive exchange can be significant and should be included in a comprehensive model.