Eric M. Valentine1, Zulfiqar Ali2 and David C. Swailes1
1 University of Newcastle upon Tyne, U.K.,
2 Technical University of Lahore, Pakistan.
Address for correspondence: Department of Civil Engineering, University of
Newcastle upon Tyne, Newcastle upon Tyne, NE1 7RU, U.K.
Tel: (44)191 222 6413, Fax: (44)191 222 6613.
E-mail: eric.valentine@ncl.ac.uk
Abstract: A one-dimensional two-zone mathematical model, comprising a pair of advection-dispersion equations coupled by a mass exchange term, is proposed to study longitudinal dispersion in channels with sequences of pools and riffles. An implicit finite-difference numerical scheme is employed, and its effectiveness is assessed with reference to known analytical solutions. Sets of longitudinal dispersion experiments were performed on various simple geometries of sequences of pools and riffles developed in a laboratory flume. A neutrally buoyant saline dispersant was used and detected with a conductivity method. The results are compared with corresponding numerical solutions to calibrate the two-zone model. The effectiveness of the numerical schemes in solving the two zone model is demonstrated.
Keywords: longitudinal dispersion, natural channels, pools, riffles, numerical model
It is widely acknowledged that simple Fickian models do not provide accurate descriptions of the longitudinal transport of contaminants in natural channels, especially during early stages of the mixing process (Young and Wallis, 1993). Thackston and Schnelle (1970), Valentine and Wood (1977, 1979a, 1979b), Bencala and Walters (1983), Chikwendu and Ojiakor (1985) and others have developed models to explain mixing processes in natural channels..
In line with earlier works, [see for example Smith (1976, 1979), Bencala and Walters (1983), Chikwendu and Ojiakor (1985), Seo (1990a, 1990b), Seo and Maxwell (1992)] and considering flow characteristics of pools and riffles (Miller and Wenzel, 1984, 1985), a one-dimensional two-zone model consisting of flow and slow zones is proposed. The two-zone model is applied to study longitudinal dispersion in channels with sequences of pools and riffles. This model is a more general form of the dead zone mechanism having advection and dispersion terms in both zones along with the dead zone parameters. The advection and dispersion terms of the slow zone simplify the calculation of dead zone parameters with varying discharges. The proposed model could be of value for channels with pools and riffles. Such channel flows are highly turbulent, with large variations in flow velocities and cross-sectional areas.
As in Fig. 1, the flow is divided into two zones: a flow zone (zone 1) and a boundary slow zone (zone 2). The flow zone is an upper region of water having larger flow velocities compared to the slow zone, which is mainly controlled by surface roughness and pools. This two-zone model was studied numerically using an implicit-finite difference method. The numerical results were assessed with reference to analytical solutions for uniform flows (Chikwendu and Ojiakor, 1985) with a constant dispersion coefficient. Computer programs were developed for the numerical and the analytical solutions of the model. The programs computed concentration variations, displacement, velocity, spatial variance and total mass in the flow and the slow zones.
To create a reliable experimental database and to validate and calibrate the proposed two-zone dispersion model, hydraulic and longitudinal dispersion experiments for a variety of flow rates were performed on four idealized geometries of six pool-riffle sequences constructed in a laboratory flume. Laboratory experiments were also performed to measure mass exchange coefficients between the flow and slow zones.
The objectives of
this research were to study longitudinal dispersion and the hydraulics of
channels with sequences of pools and riffles, to test and validate the numerical
schemes with the analytical solutions, and to calibrate the proposed model
parameters with the experimental data.
In this paper, the transport of neutrally buoyant solute material introduced into a fully developed turbulent flow in an open-channel is considered. In particular, attention is given to longitudinal transport in channels that possess bed storage zones (pools) along their length (Fig. 1). Stretches between pools (the riffles) exhibit significant surface roughness and typically have a relatively steep gradient compared to the pools.
Fig.1 Sketch of
pools and riffles in a natural channel, showing average velocity and
concentration in flow and slow zones.
This study is confined to slow zones adjoining the channel bed, but the trapping may occur due to slow zones associated with channel sides. In principle, however, the model given below, with suitable reinterpretation, could be applied to flows with side slow zones.
Denoting the cross-sectional average solute
concentration in zone
at the longitudinal spatial
co-ordinate
at time
by
, the evolution and interaction of these concentrations is modeled via two
advection-dispersion equations, coupled by a mass exchange term:
(1)
In this equation,
denotes the cross-sectional area
(m2) of the flow in zone n at the point
. Similarly
and
denote, respectively, the mean
fluid velocity (m/sec) and longitudinal dispersion coefficient (m2/sec)
in zone
. In the final term on the right-hand side of equation (1)
, and
denotes the mass exchange coefficient per unit length (m2/sec)
at the interface of two zones. Hence, the sign preceding this last term is
positive for
and negative for
.
It may be noted that the model expressed by equation (1) constitutes a generalization of the two zone mechanism and allows for the variations in hydraulic parameters in both zones, and also for advective and dispersive transport in the slow zone. If we assume average slow zone velocity to be zero and neglect the dispersion term then this two-zone model reduces to a simple one-dimensional advection-dispersion model with dead zone. This two zone mechanism also converges to a classical Fickian type advection-dispersion equation when the slow zone area diminishes. The experimental results presented below indicate that longitudinal advection could be a significant transport mechanism in the slow zones, and could be accounted for in any dispersion model.
Before using this numerical approach to study
the two-zone model for geometries of interest it is important to assess the
effectiveness of the method itself. This makes use of known analytical
solutions. These results (a sample is shown in Fig. 2) also demonstrate the
significance of the velocity ratio (
) to longitudinal dispersion. The mean travel time and temporal variance
progressively increased with the decrease in velocity ratio. In the following
section the numerical results of the model are compared to the laboratory
experimental data.
Fig.2 Analytical and numerical concentration-distance profiles for flow parameters U1 =1, U2 = 0.1, D=1, m = 0.001, A1 = 0.75, A2 = 0.25.
Laboratory experiments were carried out in a flume, which was rectangular in section, 12m long, 0.31m wide, 0.45m deep with painted steel bed and glass side walls. In these experiments neutrally buoyant sodium chloride and methanol solution was used as a tracer.
To calibrate and
validate the proposed two-zone dispersion model, and to develop an improved
understanding of hydraulics of channels with pools and riffles, laboratory
experiments were performed on four idealized geometries of six pool-riffle
sequences. Rectangular concrete blocks were used to model pools and riffles, and
the transitions between the pools and the riffles were discontinuous steps.
These geometries of pools and riffles were used as they allow investigation of
the relative importance of the various transport mechanism, and also provide a
simple system for validating the numerical results. The dimensions of these
pools and riffles were selected following a literature review (Keller and
Melhorn, 1978; Bhowmik and Demissie, 1982; Miller and Wenzel, 1985; Whittaker
and Jaeggi, 1982) and after undertaking physical observations of a small stream,
which had established sequences of pools and riffles.
Four sets of laboratory experiments were conducted on four geometries of pools and riffles respectively. Each set of experiments consisted of four series of hydraulic and dispersion studies at different flow rates. These experiments were performed primarily to investigate the effect of slow zones on longitudinal dispersion. A secondary aim was to study the hydraulics of channels with pools and riffles.
Four sets of longitudinal dispersion experiments, each comprising of series of systematic laboratory studies were performed to validate the parameters of the two-zone model. However, in this paper only example results are presented.
To assess the individual effects of pools and riffles, concentration-time profiles were recorded at intervals, corresponding to the upstream and downstream end of the riffles. The data were recorded at eleven stations, injecting a known quantity of the tracer solution at the upstream of the first riffle as a line source across the flume. A number of dispersion runs were performed using the same quantity of tracer solution with the same hydraulic conditions. Average concentration-time profiles were used to compute the statistical parameters such as mean travel time, temporal variance, skew, mean velocity and longitudinal dispersion coefficients, which were compared with the simulated values. The non-dimensional simulated exchange coefficient k* varied from 0.027 to 0.0315. These values were higher than the non-dimensional value 0.02 proposed by Valentine and Wood (1977) and lower than the values reported by Seo (1990a) and the values computed in this study for higher pool lengths. This variation emerged due to advection and dispersion terms employed in the slow zone of the two-zone model.
Various empirical relations of longitudinal
dispersion coefficients for open channel flow had already been developed
considering vertical and transverse velocity variations and turbulence (Elder,
1959; Fischer, 1967; Chikwendu and Ojiakor, 1985). However, the dispersion
coefficients employed in this two-zone model were for uniform open channel flow
in a flume with smooth bed and glass side walls. A number of numerical tests
were carried out to select an appropriate value of the dispersion coefficient.
The value was selected by fitting simulated curves to rising sides of
concentration-time profiles. The dispersion coefficient 0.002m2/sec
for both flow and slow zones produced reasonably comparable profiles to measured
data. The dimensionless values of dispersion coefficients (
) where hn is the zone
depth and U* is the friction velocity were also found
to be near to the non-dimensional dispersion coefficient proposed by Chikwendu and Ojiakor (1985).
Figure 3 shows experimental and numerical concentration-time profiles of Series 1 and 2 (Set 1). As shown in these figures the agreement between measured and numerical concentration profiles improved with distance along the flume. The simulated concentration profiles showed prolonged tails, having consistency to measured data.
Fig.3 Experimental and numerical concentration-time profiles. Pool depth = 0.06m, pool length = 1.4m, riffle length = 0.6m.
Fig.4 Experimental and numerical mean travel time. Series 1, Q = 0.002m3/s and Series 2, Q = 0.004m3/s. (Set 1)
Figure 4 shows the mean travel time and temporal variance of measured and simulated data. A stair like change in mean travel time and temporal variance showed the contribution of pools and riffles on longitudinal dispersion mechanism. The simulated curves showed a linear increase in mean travel time and temporal variance, which were consistent with the measured data.
This study has demonstrated the effectiveness of the numerical schemes to solve the two-zone model. Further, the proposed two-zone model takes into account the advection and dispersion of the both zones and therefore, improves the understanding of longitudinal transport in channels of this type. The advection and dispersion terms of the slow zone help in simplifying the calculation of dead zone parameters with varying discharges. It is worth mentioning that the boundary between the two zones is not associated with any drastic change in the properties of flows.
Acknowledgement
The Ministry of Education, Govt. of Pakistan supported the first author for the duration of this study.
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