Transverse diffusion of solutes in natural streams

Abstract

An analytical model is developed for the transverse diffusion of solutes from steady sources placed in a natural stream with steady discharge. In theoretical derivations, the steady-state convective diffusion equation for a natural stream is convertEd into an approximate equation suitable for analytical treatment by the introduction of cumulative partial discharge as an independent variable. Analytical solutions for the latter equation are obtained for point and line source injections of solutes. The validity of the model is tested using data from four natural streams, three of which are alined straight, while one has mild curves. The verification was satisfactory for the three straight channels, and the constant in Elder's formula for the transverse diffusion coefficient ranged from 0.2 to 0.3. In the curved channel, the model appears workable, but an increase of the diffusion coefficient in the downstream direction is observed. The applications to stream problems are discussed in the final section. The model may be employed for the description and prediction of thermal and other waste disposal problems, simulation studies by tracers, and the determination of transverse diffusion coefficient. Another use of the model, the estimation of mixing distances, is explained and demonstrated using the data of the Water Research Association, England.

Transverse mixing and surface heat exchange in the Waikato River: A comparison of two models

“…In curved laboratory channels E z reaches a maximum of about twice the average value in the downstream portion of a bend and a minimum of about half the average value in the upstream portion (Sayre & Yeh 1973). Over a study reach of several kilometees, however, one would expect some averaging out of localised variations and both Yotsukura & Cobb (1972) and Lau & Krishnappan (1981) found that reach-averaged E z values deduced from dye profiles only varied by a factor of 2 along natural river channels. Temperature is not a neutrally buoyant tracer and consequently the rate of transverse mixing is likely to be affected by buoyancy forces.…”

“…Intuitively it would seem, however, that during its passage along a river reach each tracer particle will sample the entire range of conditions associated with secondary currents and hence that an equilibrium of sorts will become established as is required by Taylor's analysis. There is certainly empirical evidence to support the use of Eqn 1 for describing tracer profiles in laboratory and field studies (Elder 1956;Yotsukura & Cobb 1972;Yotsukura & Sayre 1976). Thus, despite some theoretical weaknesses it is common practice to assume that transverse dispersion can be modelled using Fick's law.…”

Assessment of steady-state propane-gas tracer method for determining reaeration coefficients, Chenango River, New York

“…However, these areas are proportional to but not directly reducible to numerical values of <j>. In order to overcome this difficulty and to help estimate <J>, in particular, use was made of the stream-tube model of Yotsukura and Cobb (1972). The approach was to simulate the distribution of <J> observed at the upstream cross section and then, utilizing the same parameters, to estimate a distribution at the downstream section.…”

“…Choosing the injection streamline at r =30 S percent, the next step was to ensure that the tracer concentration was high enough for measurement at three streamlines of the upstream cross section. Assuming low values for the transverse mixing coefficient and utilizing the estimated bulk flow parameters, the two-dimensional steadystate concentration in terms of the mass distribution factor, <j>, was estimated by means of the stream-tube analytical model (Yotsukura and Cobb, 1972).…”

An assessment of steady-state propane-gas tracer method for reaeration coefficients — Cowaselon Creek, New York

“…This decision was partly based on the observation in the June 30 and July 1 tests that the transverse variations of the dye concentration-time curve and of the steady-state gas concentration were not significant at all cross sections. Even though the discharge for the October test was considerably higher than that for the summer tests and thus the distance for transverse uniform mixing was expected to be longer, an estimate based on the channel width of 10.7 m, depth of 0.46 m, velocity of 0.30 m sec" (meter per second), and shear velocity of 0.06 m sec" gave the mixing distance of 305 to 460 m from the injection site (Yotsukura and Cobb, 1972). The distance between the injection site and section 1 was 610 m, which was longer than the estimated mixing distance.…”