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Stochastic numerical analysis of anomalous longitudinal dispersion and dilution in shallow decelerating stream flows

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Abstract

The paper deals with the numerical analysis of solute dispersion and dilution within shallow stream flows which are characterized by a low gradient and undergo a downstream deceleration, near the confluence into large reservoirs or upstream from fluvial barrages. The main purpose of the study consists in investigating the effect that the size and the slope of the channel induce on solute transport in the presence of backwaters. The analysis, performed by a numerical stochastic Lagrangian approach, detects a characteristic travel time beyond which the longitudinal hydrodynamic dispersion, jointly with dilution, undergoes a drastic decline. One of the key results of the simulations is indeed represented by the possibility to estimate the large-time section-averaged concentration resorting to the classic Gaussian distribution characterized by the actual first- and second-order particle moments and, therefore, by increasing and asymptotically constant values. In addition, the assessment of dilution at cloud scale, performed through the entropic evaluation of the total volume occupied by the solute and a closed-form expression of the geometric mean of the reactor ratio, reveals that, after a transient characterized by the earlier tendency to the cross-sectional mixing, the backwater flows induce a global re-densification of the cloud and a permanent transverse non-uniformity.

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Abbreviations

B :

River width

c :

Solute concentration

c′:

Concentration deviatoric component

\( \bar{c} \) :

Depth-averaged concentration

\( \bar{c}\,^{\prime} \) :

Deviatoric component of the depth-averaged concentration

\( \bar{c}\,^{\prime}_{u} \) :

Deviatoric component of the depth-averaged concentration in uniform flow

C :

Section-averaged concentration

C u :

Uniform flow section-averaged concentration

Cpdf:

Cumulative probability density function

CV u :

Coefficient of variation of the depth-averaged concentration

E :

System entropy

E d :

Discrete case system entropy

E max :

Maximum system entropy

ε x :

Longitudinal mixing coefficient

ε xm :

Elder’s longitudinal mixing coefficient

\( \varepsilon_{y} = \bar{\varepsilon }_{y} \) :

Transverse mixing coefficient

ε z :

Vertical mixing coefficient

D Mx :

Longitudinal macro-dispersion coefficient

Φ:

Chézy coefficient

Φ0 :

Uniform flow Chézy coefficient

g :

Specific gravity

H :

Flow average depth

H 0 :

Uniform flow average depth

H C :

Critical depth

i f :

Bed slope

I xx :

Longitudinal inertia moment

I xxu :

Uniform flow longitudinal inertia moment

j :

Energy loss

k B :

Boltzmann constant

K s :

Gauckler–Strickler–Manning coefficient

λ:

Single-layer center of mass

m :

Tracer mass

M :

Reactor ratio

M g :

Geometric mean of the reactor ratio

M gu :

Uniform-flow geometric mean of the reactor ratio

p :

Probability density function of the single particle position

Ω:

Flow area

Π xu :

Uniform-flow center of mass variance

q s :

Mass flux vector

Q :

Flow rate

R :

Hydraulic radius

R 0 :

Uniform-flow hydraulic radius

s :

Coordinate measured along the macroscopic direction of the flow

σ 2 c :

Spatial variance of the depth-averaged concentration

σ 2 cu :

Uniform flow spatial variance of the depth-averaged concentration

S x :

Center of mass longitudinal coordinate

S y :

Center of mass transverse coordinate

t :

Time

T xx :

Longitudinal one-particle covariance

T xxu :

Uniform flow longitudinal one-particle covariance

u :

Longitudinal component of the velocity

u ′ :

Longitudinal velocity deviatoric component

\( \bar{u} \) :

Depth-averaged velocity

\( \bar{u}_{0} \) :

Uniform flow depth-averaged velocity

\( \bar{u}' \) :

Deviatoric component of the depth-averaged velocity

u * :

Shear velocity

u *0 :

Uniform-flow shear velocity

U :

Section-averaged velocity

U 0 :

Uniform-flow section-averaged velocity

x :

Longitudinal coordinate

X :

Particle position vector

X :

Longitudinal component of the particle position vector

X B :

Brownian component of the trajectory

v :

Transverse component of the velocity

y :

Transverse coordinate

Y :

Transverse component of the particle position vector

w :

Vertical component of the velocity

z :

Vertical coordinate

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Authors and Affiliations

  1. School of Engineering, University of Basilicata, Viale dell’Ateneo Lucano, 10, 85100, Potenza, Italy

    Marilena Pannone & Annamaria De Vincenzo

Authors
  1. Marilena Pannone
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  2. Annamaria De Vincenzo
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Correspondence to Marilena Pannone.

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Pannone, M., De Vincenzo, A. Stochastic numerical analysis of anomalous longitudinal dispersion and dilution in shallow decelerating stream flows. Stoch Environ Res Risk Assess 29, 2087–2100 (2015). https://doi.org/10.1007/s00477-014-1006-0

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