Rouse Equation Research Articles

A Co‐Spectral Budget Model Links Turbulent Eddies to Suspended Sediment Concentration in Channel Flows

AbstractThe vertical distribution of suspended sediment concentration remains a subject of active research given its relevance to a plethora of problems in hydraulics, hydrology, ecology, and water quality control. Many of the classical theories developed over the course of 90 years represent the effects of turbulence on suspended sediments (SS) using an effective mixing length or eddy diffusivity without explicitly accounting for the energetics of turbulent eddies across scales. To address this gap, the turbulent flux of sediments is derived using a co‐spectral budget (CSB) model that can be imminently used in SS and other fine particle transport models. The CSB closes the pressure‐redistribution effect using a spectral linear Rotta scheme modified to include isotropization of production and interactions between turbulent eddies and sediment grains through a modified scale‐dependent de‐correlation time. The result is a formulation similar in complexity to the widely used Rouse's equation but with all characteristic scales, Reynolds number, and Schmidt number effects derived from well‐established spectral shapes of the vertical velocity and accepted constants from turbulence models. Finally, the proposed CSB model can recover Prandtl's and Rouse's equations under restricted conditions.

The Influence of Vertical Velocity Distribution on the Calculation of Suspended Sediment Concentration

Based on the advection‐diffusion equation of suspended sediment, a general formula of vertical distribution of suspended sediment concentration was derived by considering the influence of vertical velocity. The new formula was tested against over 3000 sets of field data and obtained a reasonable agreement. Comparing with the Rouse equation of concentration, the accuracy of the new formula increases significantly, and the shortcoming of the underestimation of the Rouse profile in practical application is revised. Through the analysis of the new formula with different vertical time‐averaged velocity coefficients m, it was found that vertical velocity does have an impact on the accurate estimation of sediment concentration, and the extent of which is related to the value of sediment concentration. Utilizing the regression analysis, it was found the vertical time‐averaged velocity coefficient m increases with the height above the bed.

Hydro-Sedimentary Study of a Tidal River: Case Study of the Arvand River in Iran

Abstract The present field study investigated hydraulic and sediment in tidal conditions. Measurements were made for two stations at neap tide and spring tide for 13-hour periods at no-dimensional depths of 0.2, 0.6 and 0.8 of the water level each time. In these measurements, the parameters of velocity, direction of flow, electrical conductivity, temperature and depth were measured directly, and to determine the amount of suspended sediments per hour, 3 samples of one litre from the mentioned depths were measured by the instantaneous vertical sampler and taken to a laboratory. An examination of the depth charts of the suspended sediments concentration for all stations, and in both the neap tide and the spring tide, showed that, in general the suspended sediments concentration increased with increasing depth. Moreover, the deep distribution of sediments concentration showed that the values obtained in the laboratory corresponded well with the values calculated from the Rouse equation, and as the depth increases, the sediments concentration usually increases.

Effects of non-locality on unsteady nonequilibrium sediment transport in turbulent flows: A study using space fractional ADE with fractional divergence

This study aims to develop a mathematical model that can describe the vertical distribution of suspended sediment particles in an open channel turbulent flow under unsteady nonequilibrium condition with non-local mixing effect. A space non-local transport of sediment is considered that arises due to turbulent bursting where the hopping height of a particle is not limited to a small distance as described by classical theory of turbulence, rather it can jump upto any height. To take into account this fact unlike previous researchers who used Rouse equation, Hunt equation or traditional advection–diffusion equation as the governing equation, the present study uses fractional advection–diffusion equation (fADE) where space fractional derivative for the local diffusive flux has been used that counts the effect of non-locality. The fractional derivative is considered in Caputo sense and the order α of the fractional derivative corresponds to α-order Le´vy stable distribution with value 1<α≤2. The non-local effect has been considered in the boundary conditions as well as in the non-linear models of sediment diffusivity. Four types of sediment diffusivity are considered which are generalized non-linear models for broad applicability of the study. The fADE together with the boundary and initial conditions, is solved by Chebyshev collocation method and Euler backward method. This proposed method is unconditionally convergent and converges more rapidly than other previous methods used in similar studies. Effects of non-local mixing have been investigated for transient and bottom concentration distributions. Proposed models have been validated for sediment distribution under steady and unsteady condition with existing experimental data and satisfactory results are obtained for all choices of sediment diffusivity. The variation of transient and bottom concentration with non-local effects are physically justified. Validation results show that the model can be applied to describe vertical concentration distribution in unsteady and steady turbulent flows in practical situation.

Reduction of birefringence by dynamic asymmetry in miscible blends of dissimilar polycarbonates

We found that the birefringence of miscible blends of dissimilar polycarbonates was lower than those of the neat component polymers though the sign of the birefringence was same. The modulus and birefringence of the blends decreased more gently in the glass transition region but more steeply in the rubbery plateau region than those of the neat component polymers. The modulus and birefringence relaxation curve fitting with the KWW and modified Rouse equations revealed wider distortion relaxation and orientation distributions in the blends than in the neat component polymers. The wide distortion relaxation might be attributed to the dynamic asymmetry of the blends. The orientation relaxed faster in the blend than in the component polymers, indicating accelerated orientation relaxation, which could be explained by assuming cooperative relaxation of the component polymers strongly affected by fast relaxation component of the low-Tg one due to the dynamic asymmetry with a large glass transition temperature difference between the component polymers. Owing to the accelerated relaxation, the elongated blends exhibited lower birefringence than the component polymers.

Precise Locating Control for a Polar Crane Based on Sliding Mode Active Disturbance Rejection Control and Quadratic Programming Algorithm

A polar crane is a large-scale special lifting equipment operated in a nuclear power plant. To address the precise locating control problem of a polar crane with the center of gravity shifting, with cross-coupling, and with external disturbance, an effective control scheme is proposed in this paper. Firstly, a nonholonomic constraint dynamic model of the polar crane is established according to the Lagrange–Rouse equation. Then, an expansion state observer (ESO) of the active disturbance rejection control (ADRC) method is applied to estimate and compensate the cross-coupling disturbance in real-time. To improve the robustness and convergence speed of the control system, the nonsingular terminal sliding mode (NTSM) control method is incorporated with ADRC and the stability of the controller is proven by the Lyapunov function approach. Furthermore, to solve the problem of redundant actuation and to reduce trajectory deviation of the bridge truck, the contact forces of the horizontal guide device are introduced into the quadratic programming (QP) optimization algorithm. Finally, the effectiveness and superiority of the proposed control scheme are illustrated by simulation results.

Quantification of probabilistic concentrations for mixed-size sediment particles in open channel flow

The Rouse equation is a well-known deterministic model for suspended concentrations. However, the transport of sediment particles is influenced by several random variables, such as non-uniform sediment size and turbulence structure. Experiments have demonstrated that the stochastic characteristics of turbulence structure, such as ejections and sweeps, can cause fluctuations in sediment concentrations. A new method is proposed to quantify the probabilistic sediment concentrations. In this study, the multiple-state discrete-time Markov chain and stochastic particle tracking model were used to simulate sediment transport with spatially and temporally varying probabilistic concentrations under the stochastic turbulence structure. Point estimate methods were adopted to estimate the variability of non-uniform sediment sizes. The proposed model was implemented for three cases. In the first case, the proposed model was validated against the experimental data. In the second case, spatial and temporal concentrations at high and low Rouse numbers with mean and non-uniform sediment sizes were compared. The result demonstrates that in the prediction with mean sediment sizes, the sediment concentration is overestimated near the bed, and the advection of the sediment concentration in the x-direction is underestimated. In the last case, higher-order statistical moments of the fluctuating concentrations were estimated through simulations using the proposed model. Simulation results conducted using the proposed method were compared with experimental data. The results revealed that the prediction results based on the proposed model are in good agreement with experimental data.

Entrainment and suspension of sand and gravel

Abstract. The entrainment and suspension of sand and gravel are important for the evolution of rivers, deltas, coastal areas, and submarine fans. The prediction of a vertical profile of suspended sediment concentration typically consists of assessing (1) the concentration near the bed using an entrainment relation and (2) the upward vertical distribution of sediment in the water column. Considerable uncertainty exists in regard to both of these steps, especially the near-bed concentration. Most entrainment relations have been tested against limited grain-size-specific data, and no relations have been evaluated for gravel suspension, which can be important in bedrock and mountain rivers. To address these issues, we compiled a database with suspended sediment data from natural rivers and flume experiments, taking advantage of the increasing availability of high-resolution grain size measurements. We evaluated 12 dimensionless parameters that may determine entrainment and suspension relations and applied multivariate regression analysis. A best-fit two-parameter equation (r2=0.79) shows that near-bed entrainment, evaluated at 10 % of the flow depth, decreases with the ratio of settling velocity to skin-friction shear velocity (wsi/u∗skin), as in previous relations, and increases with Froude number (Fr), possibly due to its role in determining bedload-layer concentrations. We used the Rouse equation to predict concentration upward from the reference level and evaluated the coefficient βi, which accounts for differences in the turbulent diffusivity of sediment from the parabolic eddy viscosity model used in the Rouse derivation. The best-fit relation for βi (r2=0.40) indicates greater relative sediment diffusivities for rivers with greater flow resistance, possibly due to bedform-induced turbulence, and larger wsi/u∗skin; the latter dependence is nonlinear and therefore different from standard Rouse theory. In addition, we used empirical relations for gravel saltation to show that our relation for near-bed concentration also provides good predictions for coarse-grained sediment. The new relations extend the calibrated parameter space over a wider range in sediment sizes and flow conditions compared to previous work and result in 95 % of concentration data throughout the water column predicted within a factor of 9.

Rebuttal on “A mathematical model on depth-averaged β-factor in open-channel turbulent flow” Environmental Earth Sciences 77:253 (2018)

In a recent paper published in Environmental Earth Sciences, Jain et al. (Environ Earth Sci 77:253, 2018) proposed an interesting study about the $$\beta$$-factor (the inverse of the turbulent Schmidt number, i.e. the ratio of momentum diffusivity to mass diffusivity in a turbulent flow). They proposed an equation for the depth-averaged $$\beta$$-factor which was used in the Rouse equation to calculate concentration profiles of suspended sediments in open-channel turbulent flows. Despite the interest, the study shows a weakness related to some inconsistencies and contradictions in the method. The main result should be improved. In this note, the weakness in this study will be pointed out. Some used equations are in contradiction with the initial assumptions. The Rouse equation is a solution of the steady-state one-dimensional convection–diffusion equation with a parabolic eddy viscosity which is based on a logarithmic velocity profile. In the same study, Jain et al. (2018) used two different mixing length equations, parabolic and linear relations. The parabolic mixing length and a constant mixing velocity are in contradiction with the used exponential-type eddy viscosity profile. The used equation for coefficient $$\gamma$$ is based on the finite mixing length model and the related assumptions of a linear eddy viscosity and a logarithmic velocity distribution (Nielsen and Teakle in Phys Fluids 16(7):2342–2348, 2004) which are in contradiction with the used exponential-type eddy viscosity profile and velocity distribution. An improved approach and new equations for the depth-averaged $$\beta$$-factor are proposed. The first is based on the equation for the coefficient $$A$$ proposed by Jain et al. (2018) and a second based on a linear function for $$A$$ which gives an equation for $$\beta$$ similar to that of van Rijn (J Hydraul Eng ASCE 1104(11):1613–1641, 1984).

Turbulent diffusion modelling of sediment in turbidity currents: An experimental validation of the Rouse approach

AbstractThe margins of submarine channels are characterized by deposits that fine away from the channel thalweg. This grain‐size trend is thought to reflect upward fining trends in the currents that formed the channels. This assumption enables reconstruction of turbidity currents from the geologic record, thereby providing insights into the overall sediment load of the system. It is common to assume that the density structure of a turbidity current can be modelled with simple diffusion models, such as the Rouse equation. Yet the Rouse equation was developed to describe how particles should be distributed through the water column in open‐channel flows, which fundamentally differ from turbidity currents in terms of their flow structure. Consequently, a rigorous appraisal of the Rouse model in deep‐marine settings is needed to validate the aforementioned flow reconstructions. The present study addresses this gap in the literature by providing a robust evaluation of the Rouse model's predictions of vertical particle segregation in two experimental turbidity currents that differ only in terms of their initial bed slopes (4° versus 8°). The concentration profiles of the coarsest sediment, which is suspended predominantly in the lower part of the flow, is accurately reproduced by the Rouse equation. Significant mismatches appear, however, in the concentration of finer grained sediment, especially towards the top of the flow. This problem is caused by the mixing with clear water at the top of turbidity currents. Caution is therefore advised in applying a Rouse model to levee overspill and levee‐crest deposits. Nonetheless, the Rouse model shows good agreement with laboratory measurements in the lower regions of the flow and for the coarser grains that are predominantly transported in the lower sections of submarine channels.

Estimation of Reservoir Sediment Flux through Bottom Outlet with Combination of Numerical and Empirical Methods

Sediment deposition issues for reservoirs are important in Taiwan because the severe deposition could excessively decrease the reservoir lifecycle. Extreme storm events usually can carry a massive amount of sediment into reservoirs, and deposition will happen unless the incoming material can pass through sluice gates. When it comes with high concentration, the density current flow is prone to be generated, and the bottom outlets are the most effective sluice gate to release the sediment. In order to improve the sediment release efficiency, an accurate estimation of arriving concentration and time of the density current can be useful for the reservoir management. This study develops a two-stage approach which combines a numerical model (SRH2D) and the modified Rouse equation to predict the sediment flux of the reservoir. The numerical model was verified and applied to establish the relation between inflow and dam face concentration. The modified Rouse equation then adopted this relation to estimate the proper exponential parameter. As a result, the sediment flux amount at each bottom outlet can be accurately predicted by this equation. With this means, an early warning system can be established for reservoir operation, which can improve release efficiency during typhoons.

Experimental validation of a cohesive suspended sediment transport model for two Mexican rivers

BackgroundThe south of Mexico is the least developed part of the country but its basins generate more than 60% of the country’s available water. Though disturbances to the rivers caused by development are still low, there is concern about the environmental health of the rivers. The calculation of the sediment load with a focus in fine fraction, where nutrients are found, is a priority. However, models for suspended sediment transport in large rivers are difficult to formulate because they carry a large amount of cohesive sediments, those cohesive sediments form aggregates or flocs with primary particles that are less than 65 μm. The hydrodynamic behavior of flocs depends on their size, density and shape, which differs from that of non-cohesive sediments as theirs depends on their interaction with the water column. A classical model to calculate suspended sediment concentration profiles for steady flow conditions is the Rouse equation, which has been extensively validated for non-cohesive suspended sediment. Some authors have demonstrated that when applied in some large rivers in conjunction with non-cohesive settling velocity models it does not perform very well. The difficulty comes from the fact that most of the suspended sediment charge in large rivers is constituted by cohesive sediments.ResultsSuspended sediments from Mexico’s two largest rivers Usumacinta and Grijalva, with a mean flow rate near river mouth of 2020 m3/s and 1150 m3/s respectively, were analyzed in a rotating annular flume (RAF). The shear velocity obtained in the field by ADCP was reproduced in the annular flume, the size and shape of flocs were obtained by means of PTV. Settling velocity was also obtained to calibrate a settling velocity model appropriate for cohesive sediments.ConclusionsThe settling velocity model developed for cohesive sediments in conjunction with the Rouse equation allowed the reproduction of suspended sediment concentration profiles for the rivers Usumacinta and Grijalva. The estimated concentration profiles were compared and validated with the measured concentration profiles in the field. Thus, the model obtained through this research can be used to estimate nutrient delivery to the sea from the largest rivers in Mexico.

Suspension concentration distribution in turbulent flows: An analytical study using fractional advection–diffusion equation

In this study vertical distribution of sediment particles in steady uniform turbulent open channel flow over erodible bed is investigated using fractional advection–diffusion equation (fADE). Unlike previous investigations on fADE to investigate the suspension distribution, in this study the modified Atangana–Baleanu–Caputo fractional derivative with a non-singular and non-local kernel is employed. The proposed fADE is solved and an analytical model for finding vertical suspension distribution is obtained. The model is validated against experimental as well as field measurements of Missouri River, Mississippi River and Rio Grande conveyance channel and is compared with the Rouse equation and other fractional model found in literature. A quantitative error analysis shows that the proposed model is able to predict the vertical distribution of particles more appropriately than previous models. The validation results shows that the fractional model can be equally applied to all size of particles with an appropriate choice of the order of the fractional derivative α. It is also found that besides particle diameter, parameter α depends on the mass density of particle and shear velocity of the flow. To predict this parameter, a multivariate regression is carried out and a relation is proposed for easy application of the model. From the results for sand and plastic particles, it is found that the parameter α is more sensitive to mass density than the particle diameter. The rationality of the dependence of α on particle and flow characteristics has been justified physically.

A mathematical model on depth-averagedβ-factor in open-channel turbulent flow

The well-known Rouse equation is the most widely used equation to determine the vertical distribution of suspended sediment concentration in an open-channel flow. The exponent of Rouse equation, known as Rouse number, contains the parameter β defined by the ratio of sediment diffusion coefficient to turbulent diffusion coefficient. As such to measure sediment concentration accurately, an appropriate expression for β is essentially needed. The present study, therefore, focuses on the derivation of depth-averaged β through modified expressions of sediment and turbulent diffusion coefficients. A regression analysis is done to establish the relation between β and normalized settling velocity, and the relation is used to determine suspension concentration.

Dynamics of Rouse chains undergoing head-to-head association and dissociation: Difference between dielectric and viscoelastic relaxation

Polymer chains having type-A dipoles parallel along the chain backbone exhibit slow dielectric relaxation reflecting their global motion. For monofunctionally head-associative Rouse chains having type-A dipoles, the Rouse equation of motion was combined with the association/dissociation kinetics to calculate the dielectric relaxation function, Φj(t) with j = 1 and 2 for unimer and dimer. Φ1(t) reflects the orientation of the end-to-end vector of the unimer, whereas Φ2(t) detects the orientation of two end-to-center vectors of the dimer (having symmetrical dipole inversion), both being in the direction of the applied electric field. The calculation was made by mapping the conformation of dissociating dimer onto the created unimer and vice versa with the aid of the Rouse eigenmode expansion of the bond vector of segments (Gaussian subchains). It turned out that Φj(t) is not influenced by the association/dissociation reaction. This result makes a striking contrast to the behavior of the viscoelastic relaxation function gj(t): gj(t) is strongly affected by the motional coupling between the unimer and dimer due to the reaction [H. Watanabe et al., Macromolecules 48, 3014–3030 (2015)]. This difference emerged because the dielectric Φj(t) corresponds to the vectorial first-moment average of the segmental bond vector at time t, u(n,t) with n being the segment index, whereas the viscoelastic gj(t) corresponds to the tensorial second-moment average. Because of this difference in the averaging moment, Φj(t) is subjected to cancelation in the conformational mapping but gj(t) is not, so that the reaction effect emerges only for gj(t). The experimental data of head-carboxylated high-cis polyisoprene chains (having the type-A dipoles), confirming this difference, are also presented.

Derivation of different suspension equations in sediment-laden flow from Shannon entropy

In this study the well-known Rouse equation and Barenblatt equation for suspension concentration distribution in a sediment-laden flow is derived using Shannon entropy. Considering dimensionless suspended sediment concentration as a random variable and using principle of maximum entropy, probability density function of suspension concentration is obtained. A new and general cumulative distribution function for the flow domain is proposed which can describe specific previous forms reported in the literature. The cumulative distribution function is tested with a variety sets of experimental data and also compared with previous models. The test results ensure the superiority of the new cumulative distribution function. Further a modified form of the cumulative distribution function is discussed and used to derive the suspension model of Greimann et al. The model parameters are expressed in terms of the Rouse number to show the effectiveness of this study using entropy based approach. Finally a non-linear equation in the Rouse number has been suggested to compute it from the experimental data.

Distribución vertical de sedimentos en suspension en la zona de desembocadura del Rio Magdalena, Colombia

Vertical measurements of velocity and suspended sediment concentration obtained in two climatic seasons at the mouth of the Magdalena River, Colombia, were used to estimate 1) bed shear velocity, bed shear stress and the bed roughness length by Wall law, 2) the settling velocity, by fitting distributions Rouse and Rouse equation modified and vertical profiles of suspended sediment concentration, 3) bed load sediment transport parameter estimates were determined with the formulations of Meyer-Peter & Muller (1948), Nielsen (1992), and Ribberink (1998). From the estimated parameters, a number of Rouse significantly less than one (0.01 to 0.18), a settling velocity (0.08-3.15 mm s -1 ) much less than bed shear velocity (15-70 mm s -1 ), and high values in shear stress (1.1-5.8 Pa) were obtained. It can be said therefore that in the River Magdalena predominates sediment transport in suspension. The values of bottom sediment transport calculated with the three formulations represent lower rates to 5% of total sediment transport. The formulation of Meyer-Peter & Muller (1948) underestimate the transport of bottom sediment due to the conditions under which it was originally calibrated, while Nielsen (1992) and Ribberink (1998) formulations can generate better estimates, because they were originally calibrated with fine sediment, similar to the sediment type reported in the study area.

Derivation of Hunt equation for suspension distribution using Shannon entropy theory

In this study, the Hunt equation for computing suspension concentration in sediment-laden flows is derived using Shannon entropy theory. Considering the inverse of the void ratio as a random variable and using principle of maximum entropy, probability density function and cumulative distribution function of suspension concentration is derived. A new and more general cumulative distribution function for the flow domain is proposed which includes several specific other models of CDF reported in literature. This general form of cumulative distribution function also helps to derive the Rouse equation. The entropy based approach helps to estimate model parameters using suspension data of sediment concentration which shows the advantage of using entropy theory. Finally model parameters in the entropy based model are also expressed as functions of the Rouse number to establish a link between the parameters of the deterministic and probabilistic approaches.

Mean velocity and suspended sediment concentration profile model of turbulent shear flow with probability density function

This work purposes a general mean velocity and a suspended sediment concentration (SSC) model to express distribution at every point of the cross section of turbulent shear flow by using a probability density function method. The probability density function method was used to describe the velocity and concentration profiles interacted on directly by fluid particles in the turbulent shear flow to solve turbulent flow and avoid different dynamical mechanics. The velocity profile model was obtained by solving for the profile integral with the product of the laminar velocity and probability density, through adopting an exponential probability density function to express probability distribution of velocity alteration of a fluid particle in turbulent shear flow. An SSC profile model was also created following a method similar to the above and based on the Schmidt diffusion equation. Different velocity and SSC profiles were created while changing the parameters of the models. The models were verified by comparing the calculated results with traditional models. It was shown that the probability density function model was superior to log-law in predicting stream-wise velocity profiles in coastal currents, and the probability density function SSC profile model was superior to the Rouse equation for predicting average SSC profiles in rivers and estuaries. Outlooks for precision investigation are stated at the end of this article.

Suspension concentration due to combined wave-current flows over waveforms

This paper presents the results of an experimental study carried out in a laboratory to investigate the sediment suspension due to combined wave-current flows over asymmetric dunes placed successively at the flume. The flume base was made by fixing 12 artificial asymmetric dunes. Experiments were performed with current only and combined wave-current flows over the series of bottom-waves under different frequencies of surface wave in sediment-laden flow for a fixed Reynolds number. The ninth dune was selected in this study, because the flow reached to a quasi–steady–state after the seventh dune. The eddy viscosity along the dune was found to be modulated due to formation of vortices on the dune flanks with the superposition of surface waves, but changes in vertical suspension concentration were rather weak with respect to locations as well as wave frequencies. The conventional diffusion equation was employed to estimate the vertical concentration distribution along waveforms with modified eddy viscosity for different frequency of waves. The results obtained using the modified Rouse equation compare quite well with the experimental observations. This model can address the suspension concentration that occurs in combined wave-current sediment-laden flow over waveform structures in real situation.