Rouse Equation Research Articles

Distributions of velocity, turbulence, and suspended sediment over low-relief antidunes

Understanding the interactions between sediment transport and turbulence in a supercritical flow is useful in the study of river hydraulics and fluvial systems and the modeling of such flows in nature. Toward this end, 11 profiles of suspended-sediment concentration and fluid velocity were collected in supercritical conditions over low-relief antidunes in a recirculating laboratory flume. It was found that velocity profiles agreed well with the law of the wall, and that turbulence intensities were similar to those in clear-water flows. The classic Rouse equation was found to under-predict concentration values in the upper 60-80% of the flow depth and a slightly modified version was proposed and successfully tested against experimental data from the present study and others.

Controls on effective settling velocities of mud flocs in the Changjiang Estuary, China

AbstractTheoretical and experimental studies were undertaken to gain insight into physical parameters controlling the flocculation and settling properties of mud flocs in the Changjiang Estuary, China. The Rouse equation is applied to vertical profiles of suspended sediment concentration to determine the bulk mean settling velocity (ws) of sediment suspended in the Changjiang Estuary. Both in situ point‐sampled and acoustically measured profiles of suspended mud concentrations were fit selectively. The calculated settling velocities ws mainly ranged from 0·4 to 4·1 mm s−1 for the point‐sampled data set, and from 1·0 to 3·0 mm s−1 for the acoustically measured data set. Furthermore, the settling velocities of mud flocs increased with mean concentration (C̄) of mud flocs in suspension and were proportional to increasing bottom shear stress (τb) of tidal flow. The best equation for the field settling velocity of mud flocs in the Changjiang Estuary can be expressed by the power law: ws = mC̄n (m, 1·14–2·37; n, 0·84–1·03). It is suggested that C̄ and τb were the dominant physical parameters controlling the flocculation and ws of mud flocs in suspension. Copyright © 2004 John Wiley & Sons, Ltd.

Orientational anisotropy for Rouse eigenmodes during creep and recovery process

The Rouse model is a well established model for nonentangled polymer chains and its dynamic behavior under step strain has been fully analyzed in the literature. However, to the knowledge of the authors, no analysis has been made for the orientational anisotropy for the Rouse eigenmodes during the creep and creep recovery processes. For completeness of the analysis of the Rouse model, this anisotropy is calculated from the Rouse equation of motion. The calculation is simple and straightforward, but the result is intriguing in a sense that respective Rouse eigenmodes do not exhibit the single Voigt-type retardation. Instead, each Rouse eigenmode has a distribution in the retardation time. This behavior, reflecting the interplay among the Rouse eigenmodes of different orders under the constant stress condition, is quite different from the behavior under rate-controlled flow (where each eigenmode exhibits retardation/relaxation associated with a single characteristic time).

Conformational Change of Phenyl Ring Side Group during Stress Relaxation in Glassy Poly(styrene-co-acrylonitrile)

We investigated the stress and birefringence Δn during stress relaxation in poly(styrene-co-acrylonitrile). The Δn changed its sign from positive to negative during the stress relaxation with increasing relaxation time and temperature in the glassy region. The negative Δn was found to increase when the stress was released and the specimen was cooled to room temperature. Such characteristic Δn behavior is ascribed to three relaxation mechanisms consisting of distortion, orientation, and conformational change of the phenyl ring side group. The master curve of the Δn was nicely explained by assuming that the distortion relaxation is expressed by Kohlrausch−Williams−Watts equation, the orientation relaxation by modified Rouse equation, and the conformational change by a stretched exponential relaxation function. The curve fitting procedures of the master curves revealed that the conformational change of the side group is much slower than the phenyl ring vibration (β relaxation); i.e., the rotational angle of ...

Settling velocities of fine suspended particles in the Changjiang Estuary, China

Point-sampled suspended sediment concentration profiles (grain size largely<32 μm) were measured in August 1978, April 1990, and May 1990 in the Changjiang Estuary. They were selectively fit with the Rouse equation in order to calculate the median settling velocities w s,50 of fine suspended particles in the Changjiang Estuary, East China Sea. Calculated settling velocities ranged from 0.4 to 4.1 mm s −1. Furthermore, w s values increased with the mean concentration C ̄ of sediment in suspension. The best-fit equation for the field settling velocity of fine particles in the Changjiang Estuary can be expressed by the power law: w s=2.37 C ̄ 0.84 ( r 2<0.3).

Suspension flows in open channels; experimental study

The vertical distribution of the concentration of suspended particles is usually determined by solving the diffusion-convection equation under appropriate boundary conditions. The exponent in the resulting Rouse equation is the Rouse number, defined as z′ = vss /κu. The -value has been subject of much research, being ⋛ 1. This study focuses on the experimental determination of the depth-averaged -value, given by the ratio of the sediment, ε s , and the momentum, ε m , diffusion coefficients. These coefficients were evaluated under laboratory conditions from measurements of instantaneous velocity and concentration profiles, obtained simultaneously by using a non-intrusive sonar instrument, the Acoustic Particle Flux Profiler (APFP). In suspension flows over a movable bed without bed forms the measured -values at capacity condition are smaller than unity, < 1. However, it could also be shown that for flows over a movable bed with bed forms the -values are larger than unity, > 1; this is notably the case in natural waterways.

Case Study: Sediment Transport in Proposed Geomorphic Channel for Napa River

A model study evaluates sediment transport in a geomorphic channel proposed for restoration and flood damage reduction of an 11-km tidally influenced reach of the Napa River located in California. The model study employs the unsteady quasi-2D hydrodynamic and sediment transport model MIKE 11, the simplified marsh plain accretion model MARSH 98, and the Rouse equation to predict annual average morphological changes of the geomorphic channel. The adopted modeling approach allows for the simulation of salient sediment transport processes in a river estuary, including lateral and vertical sorting of sediments, and local flushing of fine cohesive and noncohesive sediments during flooding. Accretion rates, particularly within the marsh plain terrace of the multistage channel, are found to be within acceptable limits for project maintenance and ecosystem restoration purposes. This enhanced 1D modeling approach may offer a viable and cost-effective alternative compared to fully 2D and 3D models, with relatively less model set-up and run-time requirements.

Estimating coefficients in one-dimensional depth-averaged sediment transport model

Various coefficients in sediment transport models must be accounted for. Models based on depth-averaged equations and sediment carrying capacity formula contain some coefficients: α, k, and m. At the present, no widely acceptable method has been developed for determining the values of these coefficients. The focus of this paper is in the development of semi-theoretical formulas for estimating these coefficients such that, in practical applications, the uncertainty involved in selecting coefficients is minimized. Model verification shows that the coefficients obtained from the proposed formulas give a good simulation of the channel bed deformation. In addition, Rouse's equation for sediment concentration distribution will become solvable because the reference concentration can be determined from the derived expression for α. The simulated concentration profiles obtained by solving the Rouse's equation and α formula agree reasonably well with the measured data.Key words: depth-averaged model, sediment transport, sediment-carrying capacity.

Comparing Calculated and Observed Vertical Suspended-Sediment Distributions from a Hudson River Estuary Turbidity Maximum

Suspended-sediment concentrations calculated using a vertical suspended-sediment distribution equation were compared to observations from a field study of the lower turbidity maximum of the Hudson River Estuary. At four stations, an instrumented tripod measured vertical profiles of suspended-sediment concentration, current velocity, salinity and temperature through a tidal cycle. Bed and suspended-sediment samples were also analysed to determine inorganic sediment size distributions. Velocities were as high as 1·3ms−1, with suspended-sediment concentrations up to 2000mgl−1. When a well-defined pycnocline existed, cross-isopycnal mixing was strongly damped (based on the gradient Richardson number). Suspended-sediment profiles were calculated with a stratification-modified Rouse equation, using (1) reference concentrations measured at 20cm above the bed, (2) estimates of shear velocity based on the quadratic stress law, and (3) a constant sediment settling velocity of 0·22cms−1. Differences between mean calculated and observed total suspended load for each station were −17, 7, 14 and 58%, respectively. An uncertainty analysis revealed that the two parameterizations most likely to account for differences of this magnitude were those used for settling velocity and stratification. Best results were found when substituting a power law relationship for settling velocity based on suspended-sediment concentration. This demonstrates the improvement which a power law formulation can provide over the commonly used constant wsparameterization in fine sediment environments.

Computing near-bed solids transport in sewers and similar sediment-carrying open-channel flows

Near-bed solids (NBS) in sewers are known to convey a large part of the particulate polluting material and to introduce serious bias in sewer flow quality modelling when their existence is not properly acknowledged. It is proposed that a suspended-load approach should be used to characterize this material and the related flow pattern. A two-layer suspension model (proposed as an alternative to the classical Rouse equation) is applied to combined sewer flows displaying a NBS pattern, but also to capacity suspension flows carefully controlled under laboratory conditions. Both applications are conclusive and stress the importance of a good knowledge of particle settling velocity in ambient (turbulent) conditions. In combined sewers, the marked increase of effective shear velocity associated with rain runoff events is understood to lift the coarse settleable material which ordinarily remains in a concentrated NBS flow zone, into the upper part of the flow.

Langevin dynamics of polymeric manifolds in melts

The Martin-Siggia-Rose generating functional (MSR-GF) technique is used for treating the polymeric D-dimensional-manifold melt dynamics. The one- (test-) manifold dynamics and the collective dynamics are considered separately. The test-manifold dynamics is obtained by integrating out the melt collective variables. This is done within the dynamic random-phase approximation (RPA). The resulting effective-action functional of the test manifold is treated by making use of the self-consistent Hartree approximation. As a consequence, the generalized Rouse equation of the test manifold is derived, and its static and dynamic properties are studied. By making use the MSR-GF technique, the fluctuations around the RPA of the collective variables - mass density and response-field density - are investigated. As a result, the equations for the correlation and response functions are derived. The memory kernel can be specified for the ideal glass transition as a sum of all `water-melon' diagrams.

The Hartree approximation in dynamics of polymeric manifolds in the melt

The Martin–Siggia–Rose functional integral technique is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds. The integration over the collective variables of the melt can be simply implemented in the framework of the dynamical random phase approximation. The resulting effective action functional of the test manifold is treated by making use of the self-consistent Hartree approximation. As an outcome the generalized Rouse equation of the test manifold is derived and its static and dynamic properties are studied. It was found that the static upper critical dimension, duc=2D/(2−D), discriminates between Gaussian (or screened) and non-Gaussian regimes, whereas its dynamical counterpart, d̃uc=2duc, distinguishes between the simple Rouse and the renormalized Rouse behavior. We have argued that the Rouse mode correlation function has a stretched exponential form. The subdiffusional exponents for this regime are calculated explicitly. The special case of linear chains, D=1, shows good agreement with Monte-Carlo simulations.

Dynamics of polymeric manifolds in melts: the Hartree approximation

The Martin-Siggia-Rose functional technique and the self-consistent Hartree approximation is applied to the dynamics of a D-dimensional manifold in a melt of similar manifolds.The generalized Rouse equation is derived and its static and dynamic properties are studied. The static upper critical dimension discriminate between Gaussian and non-Gaussian regimes, whereas its dynamic counterpart discriminates between Rouse- and renormalized-Rouse behavior. The dynamic exponents are calculated explicitly. The special case of linear chains shows agreement with MD- and MC-simulations.

The relation between shear velocity and suspended sediment concentration over dunes: Fraser Estuary, Canada

Two types of dunes develop in the Fraser River Estuary, Canada. Symmetric dunes have stoss and lee sides of similar length, gentle lee-side slope angles and rounded crests, while asymmetric dunes have superimposed small dunes on their stoss sides, sharp crests and steeper lee sides. Symmetric dunes appear to be near-equilibrium features at higher flows. Asymmetric dunes are transitional between symmetric dunes and the small superimposed dunes that are in equilibrium with the lower velocity and sediment transport conditions. Spatially averaged velocity profiles over dunes consist of upper and lower log-linear segments. Upper segments of profiles reflect the total stress of the flow. The lower segment on symmetric dunes reflects skin friction from sand particles, but for asymmetric dunes it is skin friction plus form roughness from the superimposed dunes. Predictions from the Rouse equation indicate that sediment suspension is controlled by total stress for symmetric dunes, whereas for asymmetric dunes sediment suspension is related to stress associated with skin friction plus form roughness. It is the equilibrium bedform — the large symmetric dunes, and the small superimposed dunes in the asymmetric case — that controls sediment suspension over dunes.

Dynamics of a Polymer Test Chain in a Glass Forming Matrix: The Hartree Approximation

In this paper the Martin-Siggia-Rose formalism is used to derive a generalized Rouse equation for a test chain in a matrix which can undergo the glass transition. It is shown that the surrounding matrix renormalizes the static properties of the test chain. Furthermore the freezing of the different Rouse modes is investigated. This yields freezing temperatures which depend from the Rouse mode index.

Concentration Profile of Suspended Sediments in Open-Channel Flow at Steep Slopes

Experiments of mixture flow of fine sand and water are made in a steep open channel. Concentration profiles of suspended sand are measured. A two-layer model for the mixture flow is proposed; the lower layer of intergranular interactions and the upper layer of turbulent motion. Using the top of the lower layer as the reference level, the concentration profiles are calculated from the Rouse equation. The calculations are in good agreement with the experiments.

Predicting ripple roughness and sand resuspension under combined flows in a shoreface environment

Ripple measurements and flow and sediment dynamical data obtained from the shoreface of the Middle Atlantic Bight using instrumented tripods were analyzed to evaluate various predictors of ripple geometry and roughness. Ripple roughness controls on sand resuspension and suspended sediment concentration profiles under combined waves and currents were also evaluated. The limited observation of sand ripples in the field indicates that the Grant and Madsen (1982) method overestimates ripple roughness, while the Nielsen (1981) method tends to under-predict ripple roughness. A modified ripple predictor is thus proposed based on the Grant and Madsen method, and it is shown to give reasonable predictions under the present experiment conditions. This modified ripple prediction along with wave, current and suspended sediment concentration data recorded by the tripods were then brought into the combined-flow bottom boundary layer model of Grant and Madsen (1986) and the modified Rouse equation of Glenn and Grant (1987) to calculate sand resuspension coefficient γ 0 and to predict suspended sediment concentration profiles. It was found that under low-energy fair-weather conditions, sand ripples are in the equilibrium range and ripple roughness increases with the bed shear stress. This causes strong vortex activity close to seabed and thus higher resuspension coefficient. Reference concentrations are moderate due to this high resuspension coefficient, even though bed shear stresses are low. Under moderate storm conditions, ripple break off occurs and ripple roughness will decrease with bed shear stress. This reduces the vortex activity and hence sand resuspension coefficient γ 0. The combination of this moderately high bed shear stress and reduced but still moderate ripple roughness favours sand suspension and produces the highest reference concentration for the encountered experimental conditions. As bed shear stress is further increased, ripples are nearly washed out and sand resuspension coefficient is further decreased approaching the previously-suggested constant value of 1.3 × 10 −4. This corresponds with the lowest reference concentration despite the high bed shear stress. Suspended sediment concentrations predicted by the modified Rouse equation using this time variable resuspension coefficient and properly calculated bottom boundary layer parameters are reasonable compared to the field measured concentration profiles.

Design of Class-I Sedimentation Tanks

The present practice for the design of primary settling tanks is based on the overflow rate corresponding to the percentage removal using isoremoval plots. The design procedure would be more rational if it were based on the removal efficiency and the scour criterion of the deposited particles. This technical note presents closed-form equations for the removal efficiency and scour velocity of particles. The removal efficiency equation considers the size distribution of particles, and the scour velocity equation is based on the Rouse equation. The removal efficiency equation yields the minimum particle size completely removed by the sedimentation tank. Using these equations the design equations have been obtained for the length and width of the settling tank. Besides other parameters, the design equations involve the completely removed minimum particle size, the specific gravity of the sediment particles, and the kinematic viscosity of water. These equations will be useful for the design of grit chambers and settling basins undergoing class-I settling.

Single chain dynamics in a binary polymer blend

The dynamics of an individual chain in a binary polymer blend is investigated. A generalized Rouse equation takes into account the coupling to the dynamics of the surrounding polymer matrix. This generalizes Schweizer’s approach to a multicomponent polymer system. The results are applied to a symmetric binary blend. Emphasis is laid on the role of composition fluctuations, which are strongly enhanced close to the phase separation and then show critical slowing down. This has an impact on the dynamics of individual chains. As a result of the treatment, the internal chain motion is expected to be more strongly affected than the center-of-mass motion. A physical interpretation is presented such that the chains are strongly expelled from unfavorable environments. It is discussed how the results are modified if—instead of a mean-field treatment—the theory of critical phenomena is applied to describe composition fluctuations.

Dynamics of block-copolymer chains near the microphase separation transition

Dynamical properties of an individual diblock copolymer are investigated. A generalized Rouse equation is derived for polymer chains composed of several blocks differing with respect to their monomeric units. This equation takes into account explicitly the dynamics of the block copolymer melt surrounding the probe chain. In general, the structure of this generalized Rouse equation differs from the corresponding result for a multicomponent homopolymer blend. It is applied to diblock copolymers close to the microphase separation, which is characterized by strong composition fluctuations and the critical slowing down of their dynamics. Emphasis is laid on the role of composition fluctuations in the dynamics of individual diblock chains. A slowing down with respect to the center-of-mass motion as well as the internal relaxations is expected. Results for the Gaussian model and the Brazovskii theory are compared. They show significant differences.